Solve -5x^2-8+3x^2+10x-1

$$- 5 x ^ { 2 } - 8 + 3 x ^ { 2 } + 10 x - 1$$

Evaluate

$$-2x^{2}+10x-9$$

Solution Steps

Combine -5x^{2} and 3x^{2} to get -2x^{2}.

$$-2x^{2}-8+10x-1$$

Subtract 1 from -8 to get -9.

$$-2x^{2}-9+10x$$

Factor

$$-2\left(x-\frac{5-\sqrt{7}}{2}\right)\left(x-\frac{\sqrt{7}+5}{2}\right)$$

Steps Using the Quadratic Formula

Combine -5x^{2} and 3x^{2} to get -2x^{2}.

$$factor(-2x^{2}-8+10x-1)$$

Subtract 1 from -8 to get -9.

$$factor(-2x^{2}-9+10x)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-2x^{2}+10x-9=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-10±\sqrt{100+8\left(-9\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.

$$x=\frac{-10±\sqrt{100-72}}{2\left(-2\right)}$$

Add 100 to -72.

$$x=\frac{-10±\sqrt{28}}{2\left(-2\right)}$$

Take the square root of 28.

$$x=\frac{-10±2\sqrt{7}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-10±2\sqrt{7}}{-4}$$

Now solve the equation x=\frac{-10±2\sqrt{7}}{-4} when ± is plus. Add -10 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-10}{-4}$$

Divide -10+2\sqrt{7} by -4.

$$x=\frac{5-\sqrt{7}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{7}}{-4} when ± is minus. Subtract 2\sqrt{7} from -10.

$$x=\frac{-2\sqrt{7}-10}{-4}$$

Divide -10-2\sqrt{7} by -4.

$$x=\frac{\sqrt{7}+5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{7}}{2} for x_{1} and \frac{5+\sqrt{7}}{2} for x_{2}.

$$-2x^{2}+10x-9=-2\left(x-\frac{5-\sqrt{7}}{2}\right)\left(x-\frac{\sqrt{7}+5}{2}\right)$$