Solve Equations in Quadratic Form

Solve Equations in Quadratic Form

Sometimes when we factored trinomials, the trinomial did not appear to be in the ax² + bx + c form. So we factored by substitution allowing us to make it fit the ax² + bx + c form. We used the standard \(u\) for the substitution.

To factor the expression x⁴ − 4x² − 5, we noticed the variable part of the middle term is x² and its square, x⁴, is the variable part of the first term. (We know \({({x}^{2})}^{2}={x}^{4}.\)) So we let u = x² and factored.

Let \(u={x}^{2}\) and substitute.
Factor the trinomial.
Replace u with \({x}^{2}\).

Similarly, sometimes an equation is not in the ax² + bx + c = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the ax² + bx + c = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.

Notice that in the quadratic equation ax² + bx + c = 0, the middle term has a variable, x, and its square, x², is the variable part of the first term. Look for this relationship as you try to find a substitution.

Again, we will use the standard u to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form ax² + bx + c = 0, we say the original equation was of quadratic form.

The next example shows the steps for solving an equation in quadratic form.

How to Solve Equations in Quadratic Form

Try it.

Solve: \(6{x}^{4}-7{x}^{2}+2=0\)

Solution

We summarize the steps to solve an equation in quadratic form.

Example

Try it.

Solve: \({(x-2)}^{2}+7(x-2)+12=0.\)

Solution

Prepare for the substitution.
Let \(u=x-2\) and substitute.
Solve by factoring.
Replace \(u\) with \(x-2.\)
Solve for \(x.\)
Check:

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• How to solve equations in quadratic form.
  1. Identify a substitution that will put the equation in quadratic form.
  2. Rewrite the equation with the substitution to put it in quadratic form.
  3. Solve the quadratic equation for u.
  4. Substitute the original variable back into the results, using the substitution.
  5. Solve for the original variable.
  6. Check the solutions.

Solve Equations in Quadratic Form

Solve Equations in Quadratic Form

In the following exercises, solve.

Try it.

\({x}^{4}-7{x}^{2}+12=0\)

Solution

\(x=\pm \sqrt{3},x=\pm 2\)

Try it.

\({x}^{4}-9{x}^{2}+18=0\)

Try it.

\({x}^{4}-13{x}^{2}-30=0\)

Solution

\(x=\pm \sqrt{15},x=\pm \sqrt{2}i\)

Try it.

\({x}^{4}+5{x}^{2}-36=0\)

Try it.

\(2{x}^{4}-5{x}^{2}+3=0\)

Solution

\(x=\pm 1,x=\frac{\pm \sqrt{6}}{2}\)

Try it.

\(4{x}^{4}-5{x}^{2}+1=0\)

Try it.

\(2{x}^{4}-7{x}^{2}+3=0\)

Solution

\(x=\pm \sqrt{3},x=\pm \frac{\sqrt{2}}{2}\)

Try it.

\(3{x}^{4}-14{x}^{2}+8=0\)

Try it.

\({(x-3)}^{2}-5(x-3)-36=0\)

Solution

\(x=-1,x=12\)

Try it.

\({(x+2)}^{2}-3(x+2)-54=0\)

Try it.

\({(3y+2)}^{2}+(3y+2)-6=0\)

Solution

\(x=-\frac{5}{3},x=0\)

Try it.

\({(5y-1)}^{2}+3(5y-1)-28=0\)

Try it.

\({({x}^{2}+1)}^{2}-5({x}^{2}+1)+4=0\)

Solution

\(x=0,x=\pm \sqrt{3}\)

Try it.

\({({x}^{2}-4)}^{2}-4({x}^{2}-4)+3=0\)

Try it.

\(2{({x}^{2}-5)}^{2}-5({x}^{2}-5)+2=0\)

Solution

\(x=\pm \frac{\sqrt{22}}{2},x=\pm \sqrt{7}\)

Try it.

\(2{({x}^{2}-5)}^{2}-7({x}^{2}-5)+6=0\)

Try it.

\(x-\sqrt{x}-20=0\)

Solution

\(x=25\)

Try it.

\(x-8\sqrt{x}+15=0\)

Try it.

\(x+6\sqrt{x}-16=0\)

Solution

\(x=4\)

Try it.

\(x+4\sqrt{x}-21=0\)

Try it.

\(6x+\sqrt{x}-2=0\)

Solution

\(x=\frac{1}{4}\)

Try it.

\(6x+\sqrt{x}-1=0\)

Try it.

\(10x-17\sqrt{x}+3=0\)

Solution

\(x=\frac{1}{25},x=\frac{9}{4}\)

Try it.

\(12x+5\sqrt{x}-3=0\)

Try it.

\({x}^{\frac{2}{3}}+9{x}^{\frac{1}{3}}+8=0\)

Solution

\(x=-1,x=-512\)

Try it.

\({x}^{\frac{2}{3}}-3{x}^{\frac{1}{3}}=28\)

Try it.

\({x}^{\frac{2}{3}}+4{x}^{\frac{1}{3}}=12\)

Solution

\(x=8,x=-216\)

Try it.

\({x}^{\frac{2}{3}}-11{x}^{\frac{1}{3}}+30=0\)

Try it.

\(6{x}^{\frac{2}{3}}-{x}^{\frac{1}{3}}=12\)

Solution

\(x=\frac{27}{8},x=-\frac{64}{27}\)

Try it.

\(3{x}^{\frac{2}{3}}-10{x}^{\frac{1}{3}}=8\)

Try it.

\(8{x}^{\frac{2}{3}}-43{x}^{\frac{1}{3}}+15=0\)

Solution

\(x=\frac{27}{512},x=125\)

Try it.

\(20{x}^{\frac{2}{3}}-23{x}^{\frac{1}{3}}+6=0\)

Try it.

\(x-8{x}^{\frac{1}{2}}+7=0\)

Solution

\(x=1,x=49\)

Try it.

\(2x-7{x}^{\frac{1}{2}}=15\)

Try it.

\(6{x}^{-2}+13{x}^{-1}+5=0\)

Solution

\(x=-2,x=-\frac{3}{5}\)

Try it.

\(15{x}^{-2}-26{x}^{-1}+8=0\)

Try it.

\(8{x}^{-2}-2{x}^{-1}-3=0\)

Solution

\(x=-2,x=\frac{4}{3}\)

Try it.

\(15{x}^{-2}-4{x}^{-1}-4=0\)

Try it.

Explain how to recognize an equation in quadratic form.

Solution

Answers will vary.

Try it.

Explain the procedure for solving an equation in quadratic form.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Sekarang Anda Tidak ada kalkulator yang bisa menuntaskan soal ini, tetapi bagian-bagiannya dapat dihitung. Coba salah satu di bawah ini, atau ketik soal Anda sendiri.

⌨ Ketik ✎ Gambar

Masalah matematika Anda

π Pecahkan

Latihan (40)

Coba kerjakan setiap soal di kertas terlebih dahulu. Buka jawabannya untuk memeriksa; soal yang terverifikasi dapat dibuka di pemecah soal untuk melihat setiap langkah.

1. Factor by substitution: \({y}^{4}-{y}^{2}-20.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \(\left({y}^{2}+4\right)\left({y}^{2}-5\right)\)

2. Factor by substitution: \({(y-4)}^{2}+8(y-4)+15.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \((y-1)(y+1)\)

3. Simplify: ⓐ \({x}^{\frac{1}{2}}\cdot {x}^{\frac{1}{4}}\) ⓑ \({({x}^{\frac{1}{3}})}^{2}\) ⓒ \({({x}^{-1})}^{2}.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   ⓐ \({x}^{\frac{3}{4}}\); ⓑ \({x}^{\frac{2}{3}}\); ⓒ \({x}^{-2}\)

4. Solve: \(6{x}^{4}-7{x}^{2}+2=0\)

   Kerjakan langkah demi langkah Petunjuk

5. Solve: \({x}^{4}-6{x}^{2}+8=0\).

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \(x=\sqrt{2},x=\text{-}\sqrt{2},x=2,x=-2\)

6. Solve: \({x}^{4}-11{x}^{2}+28=0\).

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \(x=\sqrt{7},x=\text{-}\sqrt{7},x=2,x=-2\)

7. Solve: \({(x-2)}^{2}+7(x-2)+12=0.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   Prepare for the substitution.
   Let \(u=x-2\) and substitute.
   Solve by factoring.
   Replace \(u\) with \(x-2.\)
   Solve for \(x.\)
   Check:

8. Solve: \({(x-5)}^{2}+6(x-5)+8=0.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \(x=3,x=1\)

9. Solve: \({(y-4)}^{2}+8(y-4)+15=0.\)

   Kerjakan langkah demi langkah Petunjuk

   Tampilkan jawaban

   \(y=-1,y=1\)

10. Solve: \(x-3\sqrt{x}+2=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    The \(\sqrt{x}\) in the middle term, is squared in the first term \({(\sqrt{x})}^{2}=x.\) If we let \(u=\sqrt{x}\) and substitute, our trinomial will be in ax² + bx + c = 0 form.

    Rewrite the trinomial to prepare for the substitution.
    Let \(u=\sqrt{x}\) and substitute.
    Solve by factoring.
    Replace u with \(\sqrt{x}.\)
    Solve for x, by squaring both sides.
    Check:

11. Solve: \(x-7\sqrt{x}+12=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=9,x=16\)

12. Solve: \(x-6\sqrt{x}+8=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=4,x=16\)

13. Solve: \({x}^{\frac{2}{3}}-2{x}^{\frac{1}{3}}-24=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    The \({x}^{\frac{1}{3}}\) in the middle term is squared in the first term \({({x}^{\frac{1}{3}})}^{2}={x}^{\frac{2}{3}}.\) If we let \(u={x}^{\frac{1}{3}}\) and substitute, our trinomial will be in ax² + bx + c = 0 form.

    Rewrite the trinomial to prepare for the substitution.
    Let \(u={x}^{\frac{1}{3}}\) and substitute.
    Solve by factoring.
    Replace u with \({x}^{\frac{1}{3}}.\)
    Solve for \(x\) by cubing both sides.
    Check:

14. Solve: \({x}^{\frac{2}{3}}-5{x}^{\frac{1}{3}}-14=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=-8,x=343\)

15. Solve: \({x}^{\frac{1}{2}}-8{x}^{\frac{1}{4}}+15=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=81,x=625\)

16. Solve: \(3{x}^{-2}-7{x}^{-1}+2=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    The \({x}^{-1}\) in the middle term is squared in the first term \({({x}^{-1})}^{2}={x}^{-2}.\) If we let \(u={x}^{-1}\) and substitute, our trinomial will be in ax² + bx + c = 0 form.

    Rewrite the trinomial to prepare for the substitution.
    Let \(u={x}^{-1}\) and substitute.
    Solve by factoring.
    Replace u with \({x}^{-1}.\)
    Solve for \(x\) by taking the reciprocal since \({x}^{-1}=\frac{1}{x}.\)
    Check:

17. Solve: \(8{x}^{-2}-10{x}^{-1}+3=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\frac{4}{3}x=2\)

18. Solve: \(6{x}^{-2}-23{x}^{-1}+20=0.\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\frac{2}{5},x=\frac{3}{4}\)

19. \({x}^{4}-7{x}^{2}+12=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\pm \sqrt{3},x=\pm 2\)

20. \({x}^{4}-9{x}^{2}+18=0\)

    Kerjakan langkah demi langkah Petunjuk

21. \({x}^{4}-13{x}^{2}-30=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\pm \sqrt{15},x=\pm \sqrt{2}i\)

22. \({x}^{4}+5{x}^{2}-36=0\)

    Kerjakan langkah demi langkah Petunjuk

23. \(2{x}^{4}-5{x}^{2}+3=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\pm 1,x=\frac{\pm \sqrt{6}}{2}\)

24. \(4{x}^{4}-5{x}^{2}+1=0\)

    Kerjakan langkah demi langkah Petunjuk

25. \(2{x}^{4}-7{x}^{2}+3=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\pm \sqrt{3},x=\pm \frac{\sqrt{2}}{2}\)

26. \(3{x}^{4}-14{x}^{2}+8=0\)

    Kerjakan langkah demi langkah Petunjuk

27. \({(x-3)}^{2}-5(x-3)-36=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=-1,x=12\)

28. \({(x+2)}^{2}-3(x+2)-54=0\)

    Kerjakan langkah demi langkah Petunjuk

29. \({(3y+2)}^{2}+(3y+2)-6=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=-\frac{5}{3},x=0\)

30. \({(5y-1)}^{2}+3(5y-1)-28=0\)

    Kerjakan langkah demi langkah Petunjuk

31. \({({x}^{2}+1)}^{2}-5({x}^{2}+1)+4=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=0,x=\pm \sqrt{3}\)

32. \({({x}^{2}-4)}^{2}-4({x}^{2}-4)+3=0\)

    Kerjakan langkah demi langkah Petunjuk

33. \(2{({x}^{2}-5)}^{2}-5({x}^{2}-5)+2=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\pm \frac{\sqrt{22}}{2},x=\pm \sqrt{7}\)

34. \(2{({x}^{2}-5)}^{2}-7({x}^{2}-5)+6=0\)

    Kerjakan langkah demi langkah Petunjuk

35. \(x-\sqrt{x}-20=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=25\)

36. \(x-8\sqrt{x}+15=0\)

    Kerjakan langkah demi langkah Petunjuk

37. \(x+6\sqrt{x}-16=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=4\)

38. \(x+4\sqrt{x}-21=0\)

    Kerjakan langkah demi langkah Petunjuk

39. \(6x+\sqrt{x}-2=0\)

    Kerjakan langkah demi langkah Petunjuk

    Tampilkan jawaban

    \(x=\frac{1}{4}\)

40. \(6x+\sqrt{x}-1=0\)

    Kerjakan langkah demi langkah Petunjuk

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Daftar Masuk

Simbol yang digunakan di sini

Sentuh simbol apapun untuk definisi lengkap, gambar, dan apa setiap huruf di dalamnya berarti.

\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.iimaginary unit
i² = −1.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Bagaimana: Solve Equations in Quadratic Form

1. Solve equations in quadratic form
2. Identify a substitution that will put the equation in quadratic form.
3. Rewrite the equation with the substitution to put it in quadratic form.
4. Solve the quadratic equation for
5. Substitute the original variable back into the results, using the substitution.
6. Solve for the original variable.
7. Check the solutions.
8. How to solve equations in quadratic form.

Pertanyaan yang sering diajukan

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Bagian dari halaman ini diadaptasi dari OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Diringkas dan dijelaskan ulang di sini; kesalahan adalah tanggung jawab kami. Karena sumbernya dilisensikan untuk penggunaan non-komersial, pelajaran ini gratis, menjual apa-apa, dan dibagi di bawah lisensi yang sama.

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