23 $$ \begin{array}{l} ext { solve using the quadrat } \ ext { formulat. Round your } \ ext { answers to the neares } \ ext { thousalndth as needed. } \ -7 \mathrm{X}^{2}-21=-8 \mathrm{X}^{2}-11 \mathrm{X}\end{array} $$ Enter your solutions as a set:

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23 $$ \begin{array}{l}	ext { solve using the quadrat } \ 	ext { formulat. Round your } \ 	ext { answers to the neares } \ 	ext { thousalndth as needed. } \ -7 \mathrm{X}^{2}-21=-8 \mathrm{X}^{2}-11 \mathrm{X}\end{array} $$ Enter your solutions as a set:

UpStudy AI · Accepted Answer

We start with the equation
$$
-7x^2 - 21 = -8x^2 - 11x.
$$

**Step 1. Rearrange the equation to set it equal to zero.**

Add $$8x^2 + 11x$$ to both sides:
$$
-7x^2 + 8x^2 + 11x - 21 = 0.
$$
This simplifies to:
$$
x^2 + 11x - 21 = 0.
$$

**Step 2. Identify the coefficients.**

The quadratic is in the form
$$
ax^2 + bx + c = 0,
$$
with:
$$
a = 1,\quad b = 11,\quad c = -21.
$$

**Step 3. Use the quadratic formula.**

The quadratic formula is
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$
Substitute the coefficients:
$$
x = \frac{-11 \pm \sqrt{11^2 - 4(1)(-21)}}{2(1)}.
$$

**Step 4. Simplify the expression inside the square root.**

Calculate the discriminant:
$$
11^2 - 4(1)(-21) = 121 + 84 = 205.
$$
Thus,
$$
x = \frac{-11 \pm \sqrt{205}}{2}.
$$

**Step 5. Compute numerical approximations.**

Calculate $$\sqrt{205}$$:
$$
\sqrt{205} \approx 14.318.
$$

Now, determine the two solutions:

For the positive square root:
$$
x \approx \frac{-11 + 14.318}{2} = \frac{3.318}{2} \approx 1.659.
$$

For the negative square root:
$$
x \approx \frac{-11 - 14.318}{2} = \frac{-25.318}{2} \approx -12.659.
$$

**Step 6. Present the solution as a set.**

The solutions are:
$$
\{1.659,\ -12.659\}.
$$
The solutions are approximately $$ x \approx 1.659 $$ and $$ x \approx -12.659 $$.

23
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