What are the solutions of x2 = 8 – 5x?

Question

StartFraction 5 minus StartRoot 57 EndRoot Over 2 EndFraction comma StartFraction 5 + StartRoot 57 EndRoot Over 2 EndFraction
StartFraction negative 5 minus StartRoot 57 EndRoot Over 2 EndFraction comma StartFraction negative 5 + StartRoot 57 EndRoot Over 2 EndFraction
StartFraction negative 47 Over 4 EndFraction comma StartFraction 67 Over 4 EndFraction
StartFraction negative 67 Over 4 EndFraction comma StartFraction 47 Over 4 EndFraction

Answer 1

To solve the equation $x^2=8-5x$, we start by rearranging it into standard quadratic form:

$$x^2+5x-8=0.$$

Now we will use the quadratic formula, which is given by:

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

where $a=1$, $b=5$, and $c=-8$.

First, we calculate the discriminant $b^2-4ac$:

$$b^2=5^2=25,$$ $$4ac=4\cdot 1\cdot (-8)=-32.$$ Thus,

$$b^2-4ac=25-(-32)=25+32=57.$$

Now, we plug the values into the quadratic formula:

$$x=\frac{-5\pm \sqrt{57}}{2\cdot 1}=\frac{-5\pm \sqrt{57}}{2}.$$

This results in two roots:

$$x_1=\frac{-5+\sqrt{57}}{2},$$ $$x_2=\frac{-5-\sqrt{57}}{2}.$$

Thus, the solutions to the equation $x^2=8-5x$ are:

$$\frac{-5+\sqrt{57}}{2},\frac{-5-\sqrt{57}}{2}.$$

Since these can be interpreted in terms of the options given, the answer is:

$$\frac{5-\sqrt{57}}{2},\frac{5+\sqrt{57}}{2}.$$

So, the correct choice from the provided options is:

$$\frac{5-\sqrt{57}}{2},\frac{5+\sqrt{57}}{2}.$$