Solve -4x^2+5x+7

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

Factor

$$-4\left(x-\frac{5-\sqrt{137}}{8}\right)\left(x-\frac{\sqrt{137}+5}{8}\right)$$

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.

$$-4x^{2}+5x+7=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 7}}{2\left(-4\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-4\right)\times 7}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-5±\sqrt{25+16\times 7}}{2\left(-4\right)}$$

Multiply 16 times 7.

$$x=\frac{-5±\sqrt{25+112}}{2\left(-4\right)}$$

Add 25 to 112.

$$x=\frac{-5±\sqrt{137}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-5±\sqrt{137}}{-8}$$

Now solve the equation x=\frac{-5±\sqrt{137}}{-8} when ± is plus. Add -5 to \sqrt{137}.

$$x=\frac{\sqrt{137}-5}{-8}$$

Divide -5+\sqrt{137} by -8.

$$x=\frac{5-\sqrt{137}}{8}$$

Now solve the equation x=\frac{-5±\sqrt{137}}{-8} when ± is minus. Subtract \sqrt{137} from -5.

$$x=\frac{-\sqrt{137}-5}{-8}$$

Divide -5-\sqrt{137} by -8.

$$x=\frac{\sqrt{137}+5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{137}}{8} for x_{1} and \frac{5+\sqrt{137}}{8} for x_{2}.

$$-4x^{2}+5x+7=-4\left(x-\frac{5-\sqrt{137}}{8}\right)\left(x-\frac{\sqrt{137}+5}{8}\right)$$

Evaluate

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