Solve x[a^2+2a+2a+4]=10a^2+5a+4a-4

$$x [ a ^ { 2 } + 2 a + 2 a + 4 ] = 10 a ^ { 2 } + 5 a + 4 a - 4$$

Factor

$$10\left(a-\frac{-\sqrt{241}-9}{20}\right)\left(a-\frac{\sqrt{241}-9}{20}\right)$$

Steps Using the Quadratic Formula

Combine 5a and 4a to get 9a.

$$factor(10a^{2}+9a-4)$$

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.

$$10a^{2}+9a-4=0$$

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.

$$a=\frac{-9±\sqrt{9^{2}-4\times 10\left(-4\right)}}{2\times 10}$$

Square 9.

$$a=\frac{-9±\sqrt{81-4\times 10\left(-4\right)}}{2\times 10}$$

Multiply -4 times 10.

$$a=\frac{-9±\sqrt{81-40\left(-4\right)}}{2\times 10}$$

Multiply -40 times -4.

$$a=\frac{-9±\sqrt{81+160}}{2\times 10}$$

Add 81 to 160.

$$a=\frac{-9±\sqrt{241}}{2\times 10}$$

Multiply 2 times 10.

$$a=\frac{-9±\sqrt{241}}{20}$$

Now solve the equation a=\frac{-9±\sqrt{241}}{20} when ± is plus. Add -9 to \sqrt{241}.

$$a=\frac{\sqrt{241}-9}{20}$$

Now solve the equation a=\frac{-9±\sqrt{241}}{20} when ± is minus. Subtract \sqrt{241} from -9.

$$a=\frac{-\sqrt{241}-9}{20}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+\sqrt{241}}{20} for x_{1} and \frac{-9-\sqrt{241}}{20} for x_{2}.

$$10a^{2}+9a-4=10\left(a-\frac{\sqrt{241}-9}{20}\right)\left(a-\frac{-\sqrt{241}-9}{20}\right)$$

Evaluate

$$10a^{2}+9a-4$$

Solution Steps

Combine 5a and 4a to get 9a.

$$10a^{2}+9a-4$$