Which of the following is a solution to the quadratic equation −x2+x=−10 ? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)

Question

Responses

x=3.7
x equals 3.7

x=0.27
x equals 0.27

x=0.77
x equals 0.77

x=1.7
x equals 1.7

Answer 1

To solve the quadratic equation $-x^2+x=-10$, we can first rearrange it into standard form:

$$-x^2+x+10=0$$

Multiplying the entire equation by -1 gives us:

$$x^2-x-10=0$$

Now we can use the quadratic formula to find the solutions. The quadratic formula is given by:

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

In our equation $x^2-x-10=0$, the coefficients are:

• $a=1$
• $b=-1$
• $c=-10$

Now we substitute these values into the quadratic formula:

1. Calculate the discriminant ($b^2-4ac$): $$(-1{)}^2-4(1)(-10)=1+40=41$$

2. Calculate the two possible solutions using the quadratic formula: $$x=\frac{-(-1)\pm \sqrt{41}}{2(1)}=\frac{1\pm \sqrt{41}}{2}$$

Now we compute the two values:

$$x=\frac{1+\sqrt{41}}{2}\text{and}x=\frac{1-\sqrt{41}}{2}$$

Calculating these values, we first find $\sqrt{41}\approx 6.4$:

1. For $x=\frac{1+\sqrt{41}}{2}$: $$x\approx \frac{1+6.4}{2}\approx \frac{7.4}{2}\approx 3.7$$

2. For $x=\frac{1-\sqrt{41}}{2}$: $$x\approx \frac{1-6.4}{2}\approx \frac{-5.4}{2}\approx -2.7$$

Since the solutions to the quadratic equation are approximately $x\approx 3.7$ and $x\approx -2.7$, the only positive solution that matches the options provided is:

$$\boxed{{\displaystyle 3.7}}$$