Solve 27y ^ 4 + 15y ^ 2 = 2 | 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

$$27y^{4}+15y^{2}=2$$

Solve for y (complex solution)

$y=\frac{1}{3}\approx 0.333333333$
$y=-\frac{1}{3}\approx -0.333333333$
$y=-\frac{\sqrt{6}i}{3}\approx -0-0.816496581i$
$y=\frac{\sqrt{6}i}{3}\approx 0.816496581i$

Steps Using the Quadratic Formula

Subtract $2$ from both sides.

$$27y^{4}+15y^{2}-2=0$$

Substitute $t$ for $y^{2}$.

$$27t^{2}+15t-2=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $27$ for $a$, $15$ for $b$, and $-2$ for $c$ in the quadratic formula.

$$t=\frac{-15±\sqrt{15^{2}-4\times 27\left(-2\right)}}{2\times 27}$$

Do the calculations.

$$t=\frac{-15±21}{54}$$

Solve the equation $t=\frac{-15±21}{54}$ when $±$ is plus and when $±$ is minus.

$$t=\frac{1}{9}$$ $$t=-\frac{2}{3}$$

Since $y=t^{2}$, the solutions are obtained by evaluating $y=±\sqrt{t}$ for each $t$.

$$y=-\frac{1}{3}$$ $$y=\frac{1}{3}$$ $$y=-\frac{\sqrt{6}i}{3}$$ $$y=\frac{\sqrt{6}i}{3}$$

Solve for y

$y=-\frac{1}{3}\approx -0.333333333$
$y=\frac{1}{3}\approx 0.333333333$

Steps Using the Quadratic Formula

Subtract $2$ from both sides.

$$27y^{4}+15y^{2}-2=0$$

Substitute $t$ for $y^{2}$.

$$27t^{2}+15t-2=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $27$ for $a$, $15$ for $b$, and $-2$ for $c$ in the quadratic formula.

$$t=\frac{-15±\sqrt{15^{2}-4\times 27\left(-2\right)}}{2\times 27}$$

Do the calculations.

$$t=\frac{-15±21}{54}$$

Solve the equation $t=\frac{-15±21}{54}$ when $±$ is plus and when $±$ is minus.

$$t=\frac{1}{9}$$ $$t=-\frac{2}{3}$$

Since $y=t^{2}$, the solutions are obtained by evaluating $y=±\sqrt{t}$ for positive $t$.

$$y=\frac{1}{3}$$ $$y=-\frac{1}{3}$$