Solve -5x^2+6x+7+-x^2+4x

$$-5 { x }^{ 2 } +6x+7+- { x }^{ 2 } +4x$$

Evaluate

$$7+10x-6x^{2}$$

Solution Steps

Combine 6x and 4x to get 10x.

$$-5x^{2}+10x+7-x^{2}$$

Combine -5x^{2} and -x^{2} to get -6x^{2}.

$$-6x^{2}+10x+7$$

Factor

$$-6\left(x-\frac{5-\sqrt{67}}{6}\right)\left(x-\frac{\sqrt{67}+5}{6}\right)$$

Steps Using the Quadratic Formula

Combine 6x and 4x to get 10x.

$$factor(-5x^{2}+10x+7-x^{2})$$

Combine -5x^{2} and -x^{2} to get -6x^{2}.

$$factor(-6x^{2}+10x+7)$$

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.

$$-6x^{2}+10x+7=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-6\right)\times 7}}{2\left(-6\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-6\right)\times 7}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-10±\sqrt{100+24\times 7}}{2\left(-6\right)}$$

Multiply 24 times 7.

$$x=\frac{-10±\sqrt{100+168}}{2\left(-6\right)}$$

Add 100 to 168.

$$x=\frac{-10±\sqrt{268}}{2\left(-6\right)}$$

Take the square root of 268.

$$x=\frac{-10±2\sqrt{67}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-10±2\sqrt{67}}{-12}$$

Now solve the equation x=\frac{-10±2\sqrt{67}}{-12} when ± is plus. Add -10 to 2\sqrt{67}.

$$x=\frac{2\sqrt{67}-10}{-12}$$

Divide -10+2\sqrt{67} by -12.

$$x=\frac{5-\sqrt{67}}{6}$$

Now solve the equation x=\frac{-10±2\sqrt{67}}{-12} when ± is minus. Subtract 2\sqrt{67} from -10.

$$x=\frac{-2\sqrt{67}-10}{-12}$$

Divide -10-2\sqrt{67} by -12.

$$x=\frac{\sqrt{67}+5}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{67}}{6} for x_{1} and \frac{5+\sqrt{67}}{6} for x_{2}.

$$-6x^{2}+10x+7=-6\left(x-\frac{5-\sqrt{67}}{6}\right)\left(x-\frac{\sqrt{67}+5}{6}\right)$$