Solve (4x^2-3x+29)+(5x^2+6x-54)

$$( 4 x ^ { 2 } - 3 x + 29 ) + ( 5 x ^ { 2 } + 6 x - 54 )$$

Evaluate

$$9x^{2}+3x-25$$

Solution Steps

Combine 4x^{2} and 5x^{2} to get 9x^{2}.

$$9x^{2}-3x+29+6x-54$$

Combine -3x and 6x to get 3x.

$$9x^{2}+3x+29-54$$

Subtract 54 from 29 to get -25.

$$9x^{2}+3x-25$$

Factor

$$9\left(x-\frac{-\sqrt{101}-1}{6}\right)\left(x-\frac{\sqrt{101}-1}{6}\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and 5x^{2} to get 9x^{2}.

$$factor(9x^{2}-3x+29+6x-54)$$

Combine -3x and 6x to get 3x.

$$factor(9x^{2}+3x+29-54)$$

Subtract 54 from 29 to get -25.

$$factor(9x^{2}+3x-25)$$

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.

$$9x^{2}+3x-25=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 9\left(-25\right)}}{2\times 9}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 9\left(-25\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-3±\sqrt{9-36\left(-25\right)}}{2\times 9}$$

Multiply -36 times -25.

$$x=\frac{-3±\sqrt{9+900}}{2\times 9}$$

Add 9 to 900.

$$x=\frac{-3±\sqrt{909}}{2\times 9}$$

Take the square root of 909.

$$x=\frac{-3±3\sqrt{101}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-3±3\sqrt{101}}{18}$$

Now solve the equation x=\frac{-3±3\sqrt{101}}{18} when ± is plus. Add -3 to 3\sqrt{101}.

$$x=\frac{3\sqrt{101}-3}{18}$$

Divide -3+3\sqrt{101} by 18.

$$x=\frac{\sqrt{101}-1}{6}$$

Now solve the equation x=\frac{-3±3\sqrt{101}}{18} when ± is minus. Subtract 3\sqrt{101} from -3.

$$x=\frac{-3\sqrt{101}-3}{18}$$

Divide -3-3\sqrt{101} by 18.

$$x=\frac{-\sqrt{101}-1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{101}}{6} for x_{1} and \frac{-1-\sqrt{101}}{6} for x_{2}.

$$9x^{2}+3x-25=9\left(x-\frac{\sqrt{101}-1}{6}\right)\left(x-\frac{-\sqrt{101}-1}{6}\right)$$