Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] by factoring, let's go through the process step by step.
1. Substitute [tex]\( y = x^2 \)[/tex]:
This substitution simplifies the equation. Let [tex]\( y = x^2 \)[/tex], then the equation becomes:
[tex]\[ y^2 - 5y - 14 = 0 \][/tex]
2. Solve the quadratic equation [tex]\( y^2 - 5y - 14 = 0 \)[/tex]:
This is a standard quadratic equation. To solve it, we can use the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -14 \)[/tex]. Plugging these values into the quadratic formula gives:
[tex]\[ y = \frac{5 \pm \sqrt{25 + 56}}{2} \][/tex]
[tex]\[ y = \frac{5 \pm \sqrt{81}}{2} \][/tex]
[tex]\[ y = \frac{5 \pm 9}{2} \][/tex]
3. Calculate the solutions for [tex]\( y \)[/tex]:
There are two solutions for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{5 + 9}{2} = \frac{14}{2} = 7 \][/tex]
and
[tex]\[ y = \frac{5 - 9}{2} = \frac{-4}{2} = -2 \][/tex]
4. Back-substitute [tex]\( y = x^2 \)[/tex] to find [tex]\( x \)[/tex]:
Recall that [tex]\( y = x^2 \)[/tex]. So we have:
[tex]\[ x^2 = 7 \][/tex]
and
[tex]\[ x^2 = -2 \][/tex]
Solving these equations for [tex]\( x \)[/tex]:
- For [tex]\( x^2 = 7 \)[/tex]:
[tex]\[ x = \pm \sqrt{7} \][/tex]
- For [tex]\( x^2 = -2 \)[/tex]:
[tex]\[ x = \pm \sqrt{-2} = \pm \sqrt{2}i \][/tex]
5. Combine all solutions:
The solutions to the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] are:
[tex]\[ x = \pm \sqrt{7} \quad \text{and} \quad x = \pm \sqrt{2}i \][/tex]
So, the correct answer is:
[tex]\[ x = \pm \sqrt{7} \quad \text{and} \quad x = \pm i \sqrt{2} \][/tex]
1. Substitute [tex]\( y = x^2 \)[/tex]:
This substitution simplifies the equation. Let [tex]\( y = x^2 \)[/tex], then the equation becomes:
[tex]\[ y^2 - 5y - 14 = 0 \][/tex]
2. Solve the quadratic equation [tex]\( y^2 - 5y - 14 = 0 \)[/tex]:
This is a standard quadratic equation. To solve it, we can use the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -14 \)[/tex]. Plugging these values into the quadratic formula gives:
[tex]\[ y = \frac{5 \pm \sqrt{25 + 56}}{2} \][/tex]
[tex]\[ y = \frac{5 \pm \sqrt{81}}{2} \][/tex]
[tex]\[ y = \frac{5 \pm 9}{2} \][/tex]
3. Calculate the solutions for [tex]\( y \)[/tex]:
There are two solutions for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{5 + 9}{2} = \frac{14}{2} = 7 \][/tex]
and
[tex]\[ y = \frac{5 - 9}{2} = \frac{-4}{2} = -2 \][/tex]
4. Back-substitute [tex]\( y = x^2 \)[/tex] to find [tex]\( x \)[/tex]:
Recall that [tex]\( y = x^2 \)[/tex]. So we have:
[tex]\[ x^2 = 7 \][/tex]
and
[tex]\[ x^2 = -2 \][/tex]
Solving these equations for [tex]\( x \)[/tex]:
- For [tex]\( x^2 = 7 \)[/tex]:
[tex]\[ x = \pm \sqrt{7} \][/tex]
- For [tex]\( x^2 = -2 \)[/tex]:
[tex]\[ x = \pm \sqrt{-2} = \pm \sqrt{2}i \][/tex]
5. Combine all solutions:
The solutions to the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] are:
[tex]\[ x = \pm \sqrt{7} \quad \text{and} \quad x = \pm \sqrt{2}i \][/tex]
So, the correct answer is:
[tex]\[ x = \pm \sqrt{7} \quad \text{and} \quad x = \pm i \sqrt{2} \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.