Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to make a substitution that reduces its degree. Here's the step-by-step solution:
1. Identify the Terms:
- The given equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice that it involves terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose a Substitution:
- We want to turn the terms involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] into a quadratic form. Since [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex], a good substitution is [tex]\(u = x^2\)[/tex].
3. Substitute and Simplify:
- Substitute [tex]\(u = x^2\)[/tex] into the equation.
- Then, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] since [tex]\((x^2)^2 = u^2\)[/tex].
- The equation becomes: [tex]\(4(u^2) - 21(u) + 20 = 0\)[/tex].
4. Reformulate as a Quadratic Equation:
- The equation [tex]\(4(u^2) - 21(u) + 20 = 0\)[/tex] is now in standard quadratic form: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
By making the substitution [tex]\(u = x^2\)[/tex], we have successfully rewritten the original equation as a quadratic equation in terms of [tex]\(u\)[/tex]. Hence, the appropriate substitution to rewrite the equation is [tex]\(u = x^2\)[/tex].
1. Identify the Terms:
- The given equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice that it involves terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose a Substitution:
- We want to turn the terms involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] into a quadratic form. Since [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex], a good substitution is [tex]\(u = x^2\)[/tex].
3. Substitute and Simplify:
- Substitute [tex]\(u = x^2\)[/tex] into the equation.
- Then, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] since [tex]\((x^2)^2 = u^2\)[/tex].
- The equation becomes: [tex]\(4(u^2) - 21(u) + 20 = 0\)[/tex].
4. Reformulate as a Quadratic Equation:
- The equation [tex]\(4(u^2) - 21(u) + 20 = 0\)[/tex] is now in standard quadratic form: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
By making the substitution [tex]\(u = x^2\)[/tex], we have successfully rewritten the original equation as a quadratic equation in terms of [tex]\(u\)[/tex]. Hence, the appropriate substitution to rewrite the equation is [tex]\(u = x^2\)[/tex].
