Solve 6x^2+3x-7x+9-16

$$6 { x }^{ 2 } +3x-7x+9-16$$

Evaluate

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

Solution Steps

Combine 3x and -7x to get -4x.

$$6x^{2}-4x+9-16$$

Subtract 16 from 9 to get -7.

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

Factor

$$6\left(x-\left(-\frac{\sqrt{46}}{6}+\frac{1}{3}\right)\right)\left(x-\left(\frac{\sqrt{46}}{6}+\frac{1}{3}\right)\right)$$

Steps Using the Quadratic Formula

Combine 3x and -7x to get -4x.

$$factor(6x^{2}-4x+9-16)$$

Subtract 16 from 9 to get -7.

$$factor(6x^{2}-4x-7)$$

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.

$$6x^{2}-4x-7=0$$

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 6\left(-7\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-4\right)±\sqrt{16-24\left(-7\right)}}{2\times 6}$$

Multiply -24 times -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+168}}{2\times 6}$$

Add 16 to 168.

$$x=\frac{-\left(-4\right)±\sqrt{184}}{2\times 6}$$

Take the square root of 184.

$$x=\frac{-\left(-4\right)±2\sqrt{46}}{2\times 6}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{46}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{4±2\sqrt{46}}{12}$$

Now solve the equation x=\frac{4±2\sqrt{46}}{12} when ± is plus. Add 4 to 2\sqrt{46}.

$$x=\frac{2\sqrt{46}+4}{12}$$

Divide 4+2\sqrt{46} by 12.

$$x=\frac{\sqrt{46}}{6}+\frac{1}{3}$$

Now solve the equation x=\frac{4±2\sqrt{46}}{12} when ± is minus. Subtract 2\sqrt{46} from 4.

$$x=\frac{4-2\sqrt{46}}{12}$$

Divide 4-2\sqrt{46} by 12.

$$x=-\frac{\sqrt{46}}{6}+\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{3}+\frac{\sqrt{46}}{6} for x_{1} and \frac{1}{3}-\frac{\sqrt{46}}{6} for x_{2}.

$$6x^{2}-4x-7=6\left(x-\left(\frac{\sqrt{46}}{6}+\frac{1}{3}\right)\right)\left(x-\left(-\frac{\sqrt{46}}{6}+\frac{1}{3}\right)\right)$$