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Answer :
To solve the equation [tex]\(x^4 + 17x^2 + 70 = 0\)[/tex], we can use a substitution method. Let's break it down step-by-step:
1. Substitution: Let [tex]\(y = x^2\)[/tex]. This allows us to rewrite the equation in terms of [tex]\(y\)[/tex]:
[tex]\[
y^2 + 17y + 70 = 0
\][/tex]
2. Solve the Quadratic Equation: We now solve this quadratic equation for [tex]\(y\)[/tex] using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 17\)[/tex], and [tex]\(c = 70\)[/tex].
- Calculate the discriminant: [tex]\(b^2 - 4ac = 17^2 - 4 \times 1 \times 70 = 289 - 280 = 9\)[/tex].
- So, [tex]\(y\)[/tex] can be calculated as:
[tex]\[
y = \frac{-17 \pm \sqrt{9}}{2 \times 1} = \frac{-17 \pm 3}{2}
\][/tex]
This gives us two possible solutions for [tex]\(y\)[/tex]:
[tex]\[
y_1 = \frac{-17 + 3}{2} = -7, \quad y_2 = \frac{-17 - 3}{2} = -10
\][/tex]
3. Return to [tex]\(x\)[/tex] Terms: Recall that [tex]\(y = x^2\)[/tex], so:
- For [tex]\(y_1 = -7\)[/tex], [tex]\(x^2 = -7\)[/tex].
- For [tex]\(y_2 = -10\)[/tex], [tex]\(x^2 = -10\)[/tex].
Since we're dealing with squares equal to negative numbers, the solutions for [tex]\(x\)[/tex] are complex.
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{-7} = \pm \sqrt{7} \times i
\][/tex]
[tex]\[
x = \pm \sqrt{-10} = \pm \sqrt{10} \times i
\][/tex]
5. Write the Final Solution:
So, the solutions are:
[tex]\[-\sqrt{7}i, \sqrt{7}i, -\sqrt{10}i, \sqrt{10}i\][/tex]
These are the complex solutions to the original equation [tex]\(x^4 + 17x^2 + 70 = 0\)[/tex].
1. Substitution: Let [tex]\(y = x^2\)[/tex]. This allows us to rewrite the equation in terms of [tex]\(y\)[/tex]:
[tex]\[
y^2 + 17y + 70 = 0
\][/tex]
2. Solve the Quadratic Equation: We now solve this quadratic equation for [tex]\(y\)[/tex] using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 17\)[/tex], and [tex]\(c = 70\)[/tex].
- Calculate the discriminant: [tex]\(b^2 - 4ac = 17^2 - 4 \times 1 \times 70 = 289 - 280 = 9\)[/tex].
- So, [tex]\(y\)[/tex] can be calculated as:
[tex]\[
y = \frac{-17 \pm \sqrt{9}}{2 \times 1} = \frac{-17 \pm 3}{2}
\][/tex]
This gives us two possible solutions for [tex]\(y\)[/tex]:
[tex]\[
y_1 = \frac{-17 + 3}{2} = -7, \quad y_2 = \frac{-17 - 3}{2} = -10
\][/tex]
3. Return to [tex]\(x\)[/tex] Terms: Recall that [tex]\(y = x^2\)[/tex], so:
- For [tex]\(y_1 = -7\)[/tex], [tex]\(x^2 = -7\)[/tex].
- For [tex]\(y_2 = -10\)[/tex], [tex]\(x^2 = -10\)[/tex].
Since we're dealing with squares equal to negative numbers, the solutions for [tex]\(x\)[/tex] are complex.
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{-7} = \pm \sqrt{7} \times i
\][/tex]
[tex]\[
x = \pm \sqrt{-10} = \pm \sqrt{10} \times i
\][/tex]
5. Write the Final Solution:
So, the solutions are:
[tex]\[-\sqrt{7}i, \sqrt{7}i, -\sqrt{10}i, \sqrt{10}i\][/tex]
These are the complex solutions to the original equation [tex]\(x^4 + 17x^2 + 70 = 0\)[/tex].
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