Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To solve the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex], we will follow these steps:
1. Move all terms to one side of the equation to set it to zero:
[tex]\[ 4x^2 - 7x - 3x - 24 = 0 \][/tex]
2. Combine like terms:
[tex]\[ 4x^2 - 10x - 24 = 0 \][/tex]
3. Find the roots of the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex]. To do this, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -24\)[/tex].
- Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-10)^2 - 4(4)(-24) = 100 + 384 = 484 \][/tex]
- Compute the roots using the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm \sqrt{484}}{2 \cdot 4} = \frac{10 \pm 22}{8} \][/tex]
4. Simplify the solutions:
[tex]\[ x = \frac{10 + 22}{8} = \frac{32}{8} = 4 \][/tex]
[tex]\[ x = \frac{10 - 22}{8} = \frac{-12}{8} = -\frac{3}{2} \][/tex]
The solutions to the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex] are [tex]\(x = 4\)[/tex] and [tex]\(x = -\frac{3}{2}\)[/tex].
5. Checking which solutions apply from the given options:
- [tex]\(x = -4\)[/tex] does not apply.
- [tex]\(x = -3\)[/tex] does not apply.
- [tex]\(x = -\frac{3}{2}\)[/tex] applies.
- [tex]\(x = \frac{2}{3}\)[/tex] does not apply.
- [tex]\(x = 2\)[/tex] does not apply.
- [tex]\(x = 4\)[/tex] applies.
Therefore, the solutions to the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex] are:
- [tex]\(x = -\frac{3}{2}\)[/tex]
- [tex]\(x = 4\)[/tex]
1. Move all terms to one side of the equation to set it to zero:
[tex]\[ 4x^2 - 7x - 3x - 24 = 0 \][/tex]
2. Combine like terms:
[tex]\[ 4x^2 - 10x - 24 = 0 \][/tex]
3. Find the roots of the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex]. To do this, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -24\)[/tex].
- Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-10)^2 - 4(4)(-24) = 100 + 384 = 484 \][/tex]
- Compute the roots using the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm \sqrt{484}}{2 \cdot 4} = \frac{10 \pm 22}{8} \][/tex]
4. Simplify the solutions:
[tex]\[ x = \frac{10 + 22}{8} = \frac{32}{8} = 4 \][/tex]
[tex]\[ x = \frac{10 - 22}{8} = \frac{-12}{8} = -\frac{3}{2} \][/tex]
The solutions to the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex] are [tex]\(x = 4\)[/tex] and [tex]\(x = -\frac{3}{2}\)[/tex].
5. Checking which solutions apply from the given options:
- [tex]\(x = -4\)[/tex] does not apply.
- [tex]\(x = -3\)[/tex] does not apply.
- [tex]\(x = -\frac{3}{2}\)[/tex] applies.
- [tex]\(x = \frac{2}{3}\)[/tex] does not apply.
- [tex]\(x = 2\)[/tex] does not apply.
- [tex]\(x = 4\)[/tex] applies.
Therefore, the solutions to the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex] are:
- [tex]\(x = -\frac{3}{2}\)[/tex]
- [tex]\(x = 4\)[/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.