Solve {l}{7x^2-7x-7}{-4x^2+5x-7}

$$\left. \begin{array} { l } { 7 x ^ { 2 } - 7 x - 7 } \\ { - 4 x ^ { 2 } + 5 x - 7 } \end{array} \right.$$

Evaluate

$$3x^{2}-2x-14$$

Solution Steps

Combine 7x^{2} and -4x^{2} to get 3x^{2}.

$$3x^{2}-7x-7+5x-7$$

Combine -7x and 5x to get -2x.

$$3x^{2}-2x-7-7$$

Subtract 7 from -7 to get -14.

$$3x^{2}-2x-14$$

Factor

$$3\left(x-\frac{1-\sqrt{43}}{3}\right)\left(x-\frac{\sqrt{43}+1}{3}\right)$$

Steps Using the Quadratic Formula

Combine 7x^{2} and -4x^{2} to get 3x^{2}.

$$factor(3x^{2}-7x-7+5x-7)$$

Combine -7x and 5x to get -2x.

$$factor(3x^{2}-2x-7-7)$$

Subtract 7 from -7 to get -14.

$$factor(3x^{2}-2x-14)$$

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.

$$3x^{2}-2x-14=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\left(-14\right)}}{2\times 3}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3\left(-14\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\left(-14\right)}}{2\times 3}$$

Multiply -12 times -14.

$$x=\frac{-\left(-2\right)±\sqrt{4+168}}{2\times 3}$$

Add 4 to 168.

$$x=\frac{-\left(-2\right)±\sqrt{172}}{2\times 3}$$

Take the square root of 172.

$$x=\frac{-\left(-2\right)±2\sqrt{43}}{2\times 3}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{43}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{2±2\sqrt{43}}{6}$$

Now solve the equation x=\frac{2±2\sqrt{43}}{6} when ± is plus. Add 2 to 2\sqrt{43}.

$$x=\frac{2\sqrt{43}+2}{6}$$

Divide 2+2\sqrt{43} by 6.

$$x=\frac{\sqrt{43}+1}{3}$$

Now solve the equation x=\frac{2±2\sqrt{43}}{6} when ± is minus. Subtract 2\sqrt{43} from 2.

$$x=\frac{2-2\sqrt{43}}{6}$$

Divide 2-2\sqrt{43} by 6.

$$x=\frac{1-\sqrt{43}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{43}}{3} for x_{1} and \frac{1-\sqrt{43}}{3} for x_{2}.

$$3x^{2}-2x-14=3\left(x-\frac{\sqrt{43}+1}{3}\right)\left(x-\frac{1-\sqrt{43}}{3}\right)$$