Solve 23(x-3)^2+2x^2=8

$$23 ( x - 3 ) ^ { 2 } + 2 x ^ { 2 } = 8$$

Solve for x (complex solution)

$$x=\frac{69+\sqrt{214}i}{25}\approx 2.76+0.585149554i$$

$$x=\frac{-\sqrt{214}i+69}{25}\approx 2.76-0.585149554i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$23\left(x^{2}-6x+9\right)+2x^{2}=8$$

Use the distributive property to multiply 23 by x^{2}-6x+9.

$$23x^{2}-138x+207+2x^{2}=8$$

Combine 23x^{2} and 2x^{2} to get 25x^{2}.

$$25x^{2}-138x+207=8$$

Subtract 8 from both sides.

$$25x^{2}-138x+207-8=0$$

Subtract 8 from 207 to get 199.

$$25x^{2}-138x+199=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -138 for b, and 199 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-138\right)±\sqrt{\left(-138\right)^{2}-4\times 25\times 199}}{2\times 25}$$

Square -138.

$$x=\frac{-\left(-138\right)±\sqrt{19044-4\times 25\times 199}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-138\right)±\sqrt{19044-100\times 199}}{2\times 25}$$

Multiply -100 times 199.

$$x=\frac{-\left(-138\right)±\sqrt{19044-19900}}{2\times 25}$$

Add 19044 to -19900.

$$x=\frac{-\left(-138\right)±\sqrt{-856}}{2\times 25}$$

Take the square root of -856.

$$x=\frac{-\left(-138\right)±2\sqrt{214}i}{2\times 25}$$

The opposite of -138 is 138.

$$x=\frac{138±2\sqrt{214}i}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{138±2\sqrt{214}i}{50}$$

Now solve the equation x=\frac{138±2\sqrt{214}i}{50} when ± is plus. Add 138 to 2i\sqrt{214}.

$$x=\frac{138+2\sqrt{214}i}{50}$$

Divide 138+2i\sqrt{214} by 50.

$$x=\frac{69+\sqrt{214}i}{25}$$

Now solve the equation x=\frac{138±2\sqrt{214}i}{50} when ± is minus. Subtract 2i\sqrt{214} from 138.

$$x=\frac{-2\sqrt{214}i+138}{50}$$

Divide 138-2i\sqrt{214} by 50.

$$x=\frac{-\sqrt{214}i+69}{25}$$

The equation is now solved.

$$x=\frac{69+\sqrt{214}i}{25} x=\frac{-\sqrt{214}i+69}{25}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$23\left(x^{2}-6x+9\right)+2x^{2}=8$$

Use the distributive property to multiply 23 by x^{2}-6x+9.

$$23x^{2}-138x+207+2x^{2}=8$$

Combine 23x^{2} and 2x^{2} to get 25x^{2}.

$$25x^{2}-138x+207=8$$

Subtract 207 from both sides.

$$25x^{2}-138x=8-207$$

Subtract 207 from 8 to get -199.

$$25x^{2}-138x=-199$$

Divide both sides by 25.

$$\frac{25x^{2}-138x}{25}=-\frac{199}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}-\frac{138}{25}x=-\frac{199}{25}$$

Divide -\frac{138}{25}, the coefficient of the x term, by 2 to get -\frac{69}{25}. Then add the square of -\frac{69}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{138}{25}x+\left(-\frac{69}{25}\right)^{2}=-\frac{199}{25}+\left(-\frac{69}{25}\right)^{2}$$

Square -\frac{69}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{138}{25}x+\frac{4761}{625}=-\frac{199}{25}+\frac{4761}{625}$$

Add -\frac{199}{25} to \frac{4761}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{138}{25}x+\frac{4761}{625}=-\frac{214}{625}$$

Factor x^{2}-\frac{138}{25}x+\frac{4761}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{69}{25}\right)^{2}=-\frac{214}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{69}{25}\right)^{2}}=\sqrt{-\frac{214}{625}}$$

Simplify.

$$x-\frac{69}{25}=\frac{\sqrt{214}i}{25} x-\frac{69}{25}=-\frac{\sqrt{214}i}{25}$$

Add \frac{69}{25} to both sides of the equation.

$$x=\frac{69+\sqrt{214}i}{25} x=\frac{-\sqrt{214}i+69}{25}$$