Solve the equation 3, x, squared, minus, 8, x, minus, 23, equals, minus, 4, x, minus, 93x

Question

2
−8x−23=−4x−9 to the nearest tenth.

Answer 1

To solve the equation $3x^2-8x-23=-4x-9$, let's first rearrange all terms to one side of the equation.

Starting with the original equation:

$$3x^2-8x-23+4x+9=0$$

Combine like terms:

$$3x^2-4x-14=0$$

Now, we will use the quadratic formula to solve for $x$:

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

where $a=3$, $b=-4$, and $c=-14$.

First, calculate the discriminant:

$$b^2-4ac=(-4{)}^2-4(3)(-14)=16+168=184$$

Now, apply the quadratic formula:

$$x=\frac{-(-4)\pm \sqrt{184}}{2\cdot 3}=\frac{4\pm \sqrt{184}}{6}$$

Next, simplify $\sqrt{184}$:

$$\sqrt{184}=\sqrt{16\cdot 11}=4\sqrt{11}$$

Now substitute back into the formula:

$$x=\frac{4\pm 4\sqrt{11}}{6}=\frac{2\pm 2\sqrt{11}}{3}$$

This gives us two expressions for $x$:

$$x_1=\frac{2+2\sqrt{11}}{3}$$

$$x_2=\frac{2-2\sqrt{11}}{3}$$

Now, we need to compute these values to the nearest tenth. First, let's evaluate $2\sqrt{11}$:

$$\sqrt{11}\approx 3.3166\Rightarrow 2\sqrt{11}\approx 6.6332$$

Now calculate the two values:

1. For $x_1$:

$$x_1\approx \frac{2+6.6332}{3}=\frac{8.6332}{3}\approx 2.8777\Rightarrow x_1\approx 2.9\text{ (to the nearest tenth)}$$

2. For $x_2$:

$$x_2\approx \frac{2-6.6332}{3}=\frac{-4.6332}{3}\approx -1.5444\Rightarrow x_2\approx -1.5\text{ (to the nearest tenth)}$$

Thus, the solutions to the equation $3x^2-8x-23=-4x-9$ are:

$$x\approx 2.9\text{and}x\approx -1.5$$