Which of the following is a solution to the quadratic equation 2x2−9x=4

Question

? Assume that the solution has been rounded to the nearest hundredth. if applicable.(1 point)
Responses

x=−0.91
x equals negative 0.91

x=4.91
x equals 4.91

x=−0.77
x equals negative 0.77

x=4.77

Answer 1

To solve the quadratic equation $2x^2-9x=4$, we first need to rearrange it into standard form:

$$2x^2-9x-4=0.$$

Next, we can use the quadratic formula to find the solutions, where $a=2$, $b=-9$, and $c=-4$. The quadratic formula is given by:

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

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

$$b^2=(-9{)}^2=81,$$

$$4ac=4\times 2\times (-4)=-32,$$

$$b^2-4ac=81-(-32)=81+32=113.$$

Since the discriminant is positive, there will be two real solutions. Now we find the roots:

$$x=\frac{-(-9)\pm \sqrt{113}}{2\times 2}=\frac{9\pm \sqrt{113}}{4}.$$

Next, we estimate $\sqrt{113}$. Since ${10}^2=100$ and ${11}^2=121$, we know that $\sqrt{113}$ is between 10 and 11. A more precise estimate gives approximately $10.630$. Thus,

$$x\approx \frac{9\pm 10.630}{4}.$$

Calculating the two possible values of $x$:

1. For $9+10.630$:

$$x_1\approx \frac{19.630}{4}\approx 4.9075(roundedto4.91)$$

2. For $9-10.630$:

$$x_2\approx \frac{-1.630}{4}\approx -0.4075(roundedto-0.41)$$

However, we see that we need to report the answers rounded to the nearest hundredth as needed.

Now rounding our results:

• $x_1\approx 4.91$
• $x_2\approx -0.41$

Given the provided responses, the one that matches is $x=4.91$.

Therefore, the correct answer is:

x = 4.91.