Solve (15x)^(2)-16x+15=0 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$${ \left(15x \right) }^{ 2 } -16x+15=0$$

Solve for x (complex solution)

$x=\frac{8+\sqrt{3311}i}{225}\approx 0.035555556+0.255739065i$
$x=\frac{-\sqrt{3311}i+8}{225}\approx 0.035555556-0.255739065i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Expand $\left(15x\right)^{2}$.

$$15^{2}x^{2}-16x+15=0$$

Calculate $15$ to the power of $2$ and get $225$.

$$225x^{2}-16x+15=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $225$ for $a$, $-16$ for $b$, and $15$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 225\times 15}}{2\times 225}$$

Square $-16$.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 225\times 15}}{2\times 225}$$

Multiply $-4$ times $225$.

$$x=\frac{-\left(-16\right)±\sqrt{256-900\times 15}}{2\times 225}$$

Multiply $-900$ times $15$.

$$x=\frac{-\left(-16\right)±\sqrt{256-13500}}{2\times 225}$$

Add $256$ to $-13500$.

$$x=\frac{-\left(-16\right)±\sqrt{-13244}}{2\times 225}$$

Take the square root of $-13244$.

$$x=\frac{-\left(-16\right)±2\sqrt{3311}i}{2\times 225}$$

The opposite of $-16$ is $16$.

$$x=\frac{16±2\sqrt{3311}i}{2\times 225}$$

Multiply $2$ times $225$.

$$x=\frac{16±2\sqrt{3311}i}{450}$$

Now solve the equation $x=\frac{16±2\sqrt{3311}i}{450}$ when $±$ is plus. Add $16$ to $2i\sqrt{3311}$.

$$x=\frac{16+2\sqrt{3311}i}{450}$$

Divide $16+2i\sqrt{3311}$ by $450$.

$$x=\frac{8+\sqrt{3311}i}{225}$$

Now solve the equation $x=\frac{16±2\sqrt{3311}i}{450}$ when $±$ is minus. Subtract $2i\sqrt{3311}$ from $16$.

$$x=\frac{-2\sqrt{3311}i+16}{450}$$

Divide $16-2i\sqrt{3311}$ by $450$.

$$x=\frac{-\sqrt{3311}i+8}{225}$$

The equation is now solved.

$$x=\frac{8+\sqrt{3311}i}{225}$$ $$x=\frac{-\sqrt{3311}i+8}{225}$$

Steps for Completing the Square

Expand $\left(15x\right)^{2}$.

$$15^{2}x^{2}-16x+15=0$$

Calculate $15$ to the power of $2$ and get $225$.

$$225x^{2}-16x+15=0$$

Subtract $15$ from both sides. Anything subtracted from zero gives its negation.

$$225x^{2}-16x=-15$$

Divide both sides by $225$.

$$\frac{225x^{2}-16x}{225}=-\frac{15}{225}$$

Dividing by $225$ undoes the multiplication by $225$.

$$x^{2}-\frac{16}{225}x=-\frac{15}{225}$$

Reduce the fraction $\frac{-15}{225}$ to lowest terms by extracting and canceling out $15$.

$$x^{2}-\frac{16}{225}x=-\frac{1}{15}$$

Divide $-\frac{16}{225}$, the coefficient of the $x$ term, by $2$ to get $-\frac{8}{225}$. Then add the square of $-\frac{8}{225}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{225}x+\left(-\frac{8}{225}\right)^{2}=-\frac{1}{15}+\left(-\frac{8}{225}\right)^{2}$$

Square $-\frac{8}{225}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{225}x+\frac{64}{50625}=-\frac{1}{15}+\frac{64}{50625}$$

Add $-\frac{1}{15}$ to $\frac{64}{50625}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{225}x+\frac{64}{50625}=-\frac{3311}{50625}$$

Factor $x^{2}-\frac{16}{225}x+\frac{64}{50625}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{8}{225}\right)^{2}=-\frac{3311}{50625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{225}\right)^{2}}=\sqrt{-\frac{3311}{50625}}$$

Simplify.

$$x-\frac{8}{225}=\frac{\sqrt{3311}i}{225}$$ $$x-\frac{8}{225}=-\frac{\sqrt{3311}i}{225}$$

Add $\frac{8}{225}$ to both sides of the equation.

$$x=\frac{8+\sqrt{3311}i}{225}$$ $$x=\frac{-\sqrt{3311}i+8}{225}$$