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Answer :
To solve the equation [tex]\(2x^4 - 24x^2 + 70 = 0\)[/tex], we can use substitution to make the equation easier to handle. Here’s a step-by-step explanation:
1. Substitution: First, notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We can simplify this by introducing a substitution. Let [tex]\(y = x^2\)[/tex]. Then, [tex]\(x^4 = y^2\)[/tex]. Now, substitute these into the equation:
[tex]\[
2y^2 - 24y + 70 = 0
\][/tex]
2. Form a quadratic equation: The equation in terms of [tex]\(y\)[/tex] is now a quadratic equation. It looks like this:
[tex]\[
2y^2 - 24y + 70 = 0
\][/tex]
3. Solve the quadratic equation: We can solve the quadratic equation using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation, [tex]\(a = 2\)[/tex], [tex]\(b = -24\)[/tex], and [tex]\(c = 70\)[/tex].
4. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-24)^2 - 4 \times 2 \times 70 = 576 - 560 = 16
\][/tex]
5. Find the roots of the quadratic equation:
[tex]\[
y = \frac{24 \pm \sqrt{16}}{4}
\][/tex]
This gives us:
[tex]\[
y = \frac{24 \pm 4}{4}
\][/tex]
So, the solutions are [tex]\(y = \frac{28}{4} = 7\)[/tex] and [tex]\(y = \frac{20}{4} = 5\)[/tex].
6. Substitute back for [tex]\(x^2\)[/tex]: Remember that [tex]\(y = x^2\)[/tex], so we have [tex]\(x^2 = 7\)[/tex] and [tex]\(x^2 = 5\)[/tex].
7. Solve for [tex]\(x\)[/tex]:
- If [tex]\(x^2 = 7\)[/tex], then [tex]\(x = \pm \sqrt{7}\)[/tex].
- If [tex]\(x^2 = 5\)[/tex], then [tex]\(x = \pm \sqrt{5}\)[/tex].
8. List all solutions: Thus, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = -\sqrt{5}, \sqrt{5}, -\sqrt{7}, \sqrt{7}
\][/tex]
These are the solutions to the equation [tex]\(2x^4 - 24x^2 + 70 = 0\)[/tex], listed as [tex]\(-\sqrt{5}, \sqrt{5}, -\sqrt{7}, \sqrt{7}\)[/tex].
1. Substitution: First, notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We can simplify this by introducing a substitution. Let [tex]\(y = x^2\)[/tex]. Then, [tex]\(x^4 = y^2\)[/tex]. Now, substitute these into the equation:
[tex]\[
2y^2 - 24y + 70 = 0
\][/tex]
2. Form a quadratic equation: The equation in terms of [tex]\(y\)[/tex] is now a quadratic equation. It looks like this:
[tex]\[
2y^2 - 24y + 70 = 0
\][/tex]
3. Solve the quadratic equation: We can solve the quadratic equation using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation, [tex]\(a = 2\)[/tex], [tex]\(b = -24\)[/tex], and [tex]\(c = 70\)[/tex].
4. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-24)^2 - 4 \times 2 \times 70 = 576 - 560 = 16
\][/tex]
5. Find the roots of the quadratic equation:
[tex]\[
y = \frac{24 \pm \sqrt{16}}{4}
\][/tex]
This gives us:
[tex]\[
y = \frac{24 \pm 4}{4}
\][/tex]
So, the solutions are [tex]\(y = \frac{28}{4} = 7\)[/tex] and [tex]\(y = \frac{20}{4} = 5\)[/tex].
6. Substitute back for [tex]\(x^2\)[/tex]: Remember that [tex]\(y = x^2\)[/tex], so we have [tex]\(x^2 = 7\)[/tex] and [tex]\(x^2 = 5\)[/tex].
7. Solve for [tex]\(x\)[/tex]:
- If [tex]\(x^2 = 7\)[/tex], then [tex]\(x = \pm \sqrt{7}\)[/tex].
- If [tex]\(x^2 = 5\)[/tex], then [tex]\(x = \pm \sqrt{5}\)[/tex].
8. List all solutions: Thus, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = -\sqrt{5}, \sqrt{5}, -\sqrt{7}, \sqrt{7}
\][/tex]
These are the solutions to the equation [tex]\(2x^4 - 24x^2 + 70 = 0\)[/tex], listed as [tex]\(-\sqrt{5}, \sqrt{5}, -\sqrt{7}, \sqrt{7}\)[/tex].
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