Question

solve using the quadratic formula: $-x^2 + 4x = 13$; $-2pm3i$; $-2pm isqrt{17}$; $2pm3i$; $2pm isqrt{17}$

Explanation:

Step1: Rewrite to standard form

$-x^2 + 4x - 13 = 0$ or $x^2 - 4x + 13 = 0$

Step2: Identify a, b, c

$a=1$, $b=-4$, $c=13$

Step3: Apply quadratic formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step4: Compute discriminant

$b^2 - 4ac = (-4)^2 - 4(1)(13) = 16 - 52 = -36$

Step5: Simplify the root

$x = \frac{4 \pm \sqrt{-36}}{2} = \frac{4 \pm 6i}{2} = 2 \pm 3i$
Wait, correction: If using original $-x^2+4x-13=0$, $a=-1$, $b=4$, $c=-13$
$x = \frac{-4 \pm \sqrt{16 - 4(-1)(-13)}}{2(-1)} = \frac{-4 \pm \sqrt{16-52}}{-2} = \frac{-4 \pm \sqrt{-36}}{-2} = \frac{-4 \pm 6i}{-2} = 2 \mp 3i$ which is same as $2 \pm 3i$. Wait no, $\sqrt{-36}=6i$, so $\frac{-4 \pm 6i}{-2} = 2 \mp 3i$, which is equivalent to $2 \pm 3i$. But wait the options have $2 \pm i\sqrt{17}$? No, wait my mistake: $4ac$ for $-x^2+4x-13=0$ is $4*(-1)*(-13)=52$, $b^2=16$, $16-52=-36$, $\sqrt{-36}=6i$. So $\frac{-4 \pm 6i}{-2}=2 \mp 3i=2 \pm 3i$. So the correct option is $2 \pm 3i$.

Corrected Answer:

$2 \pm 3i$

Step1: Rewrite to standard quadratic form

$-x^2 + 4x - 13 = 0$

Step2: Define coefficients a, b, c

$a=-1$, $b=4$, $c=-13$

Step3: Substitute into quadratic formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step4: Calculate discriminant

$b^2 - 4ac = 4^2 - 4(-1)(-13) = 16 - 52 = -36$

Step5: Simplify to find roots

$x = \frac{-4 \pm \sqrt{-36}}{2(-1)} = \frac{-4 \pm 6i}{-2} = 2 \pm 3i$

Answer:

$2 \pm i\sqrt{17}$