Solve 4x^2-7x-3

$$4 { x }^{ 2 } -7x-3$$

Factor

$$4\left(x-\frac{7-\sqrt{97}}{8}\right)\left(x-\frac{\sqrt{97}+7}{8}\right)$$

Steps Using the Quadratic Formula

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.

$$4x^{2}-7x-3=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-3\right)}}{2\times 4}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 4\left(-3\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-16\left(-3\right)}}{2\times 4}$$

Multiply -16 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+48}}{2\times 4}$$

Add 49 to 48.

$$x=\frac{-\left(-7\right)±\sqrt{97}}{2\times 4}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{97}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{7±\sqrt{97}}{8}$$

Now solve the equation x=\frac{7±\sqrt{97}}{8} when ± is plus. Add 7 to \sqrt{97}.

$$x=\frac{\sqrt{97}+7}{8}$$

Now solve the equation x=\frac{7±\sqrt{97}}{8} when ± is minus. Subtract \sqrt{97} from 7.

$$x=\frac{7-\sqrt{97}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{97}}{8} for x_{1} and \frac{7-\sqrt{97}}{8} for x_{2}.

$$4x^{2}-7x-3=4\left(x-\frac{\sqrt{97}+7}{8}\right)\left(x-\frac{7-\sqrt{97}}{8}\right)$$

Evaluate

$$4x^{2}-7x-3$$