Solve x^2-4x-5quadx^2-25quadx^2+10x+25

$$x ^ { 2 } - 4 x - 5 \quad x ^ { 2 } - 25 \quad x ^ { 2 } + 10 x + 25$$

Evaluate

$$25+6x-29x^{2}$$

Solution Steps

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-4x-25x^{2}+10x+25$$

Combine -4x^{2} and -25x^{2} to get -29x^{2}.

$$-29x^{2}-4x+10x+25$$

Combine -4x and 10x to get 6x.

$$-29x^{2}+6x+25$$

Factor

$$-29\left(x-\frac{3-\sqrt{734}}{29}\right)\left(x-\frac{\sqrt{734}+3}{29}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$factor(-4x^{2}-4x-25x^{2}+10x+25)$$

Combine -4x^{2} and -25x^{2} to get -29x^{2}.

$$factor(-29x^{2}-4x+10x+25)$$

Combine -4x and 10x to get 6x.

$$factor(-29x^{2}+6x+25)$$

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.

$$-29x^{2}+6x+25=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-29\right)\times 25}}{2\left(-29\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-29\right)\times 25}}{2\left(-29\right)}$$

Multiply -4 times -29.

$$x=\frac{-6±\sqrt{36+116\times 25}}{2\left(-29\right)}$$

Multiply 116 times 25.

$$x=\frac{-6±\sqrt{36+2900}}{2\left(-29\right)}$$

Add 36 to 2900.

$$x=\frac{-6±\sqrt{2936}}{2\left(-29\right)}$$

Take the square root of 2936.

$$x=\frac{-6±2\sqrt{734}}{2\left(-29\right)}$$

Multiply 2 times -29.

$$x=\frac{-6±2\sqrt{734}}{-58}$$

Now solve the equation x=\frac{-6±2\sqrt{734}}{-58} when ± is plus. Add -6 to 2\sqrt{734}.

$$x=\frac{2\sqrt{734}-6}{-58}$$

Divide -6+2\sqrt{734} by -58.

$$x=\frac{3-\sqrt{734}}{29}$$

Now solve the equation x=\frac{-6±2\sqrt{734}}{-58} when ± is minus. Subtract 2\sqrt{734} from -6.

$$x=\frac{-2\sqrt{734}-6}{-58}$$

Divide -6-2\sqrt{734} by -58.

$$x=\frac{\sqrt{734}+3}{29}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{734}}{29} for x_{1} and \frac{3+\sqrt{734}}{29} for x_{2}.

$$-29x^{2}+6x+25=-29\left(x-\frac{3-\sqrt{734}}{29}\right)\left(x-\frac{\sqrt{734}+3}{29}\right)$$