Solve e 2x ^ 2 - 7x + 8 = 0 3x se the discriminant to determine the nature of the roots of 2 - 5x = 4/x | 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

$$2 - 5 x = 0 - \frac { 4 } { x }$$

Solve for x

$x = \frac{\sqrt{21} + 1}{5} \approx 1.116515139$
$x=\frac{1-\sqrt{21}}{5}\approx -0.716515139$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $x$.

$$x\times 2-5xx=0-4$$

Multiply $x$ and $x$ to get $x^{2}$.

$$x\times 2-5x^{2}=0-4$$

Subtract $4$ from $0$ to get $-4$.

$$x\times 2-5x^{2}=-4$$

Add $4$ to both sides.

$$x\times 2-5x^{2}+4=0$$

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.

$$-5x^{2}+2x+4=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-5$ for $a$, $2$ for $b$, and $4$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Square $2$.

$$x=\frac{-2±\sqrt{4-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Multiply $-4$ times $-5$.

$$x=\frac{-2±\sqrt{4+20\times 4}}{2\left(-5\right)}$$

Multiply $20$ times $4$.

$$x=\frac{-2±\sqrt{4+80}}{2\left(-5\right)}$$

Add $4$ to $80$.

$$x=\frac{-2±\sqrt{84}}{2\left(-5\right)}$$

Take the square root of $84$.

$$x=\frac{-2±2\sqrt{21}}{2\left(-5\right)}$$

Multiply $2$ times $-5$.

$$x=\frac{-2±2\sqrt{21}}{-10}$$

Now solve the equation $x=\frac{-2±2\sqrt{21}}{-10}$ when $±$ is plus. Add $-2$ to $2\sqrt{21}$.

$$x=\frac{2\sqrt{21}-2}{-10}$$

Divide $-2+2\sqrt{21}$ by $-10$.

$$x=\frac{1-\sqrt{21}}{5}$$

Now solve the equation $x=\frac{-2±2\sqrt{21}}{-10}$ when $±$ is minus. Subtract $2\sqrt{21}$ from $-2$.

$$x=\frac{-2\sqrt{21}-2}{-10}$$

Divide $-2-2\sqrt{21}$ by $-10$.

$$x=\frac{\sqrt{21}+1}{5}$$

The equation is now solved.

$$x=\frac{1-\sqrt{21}}{5}$$ $$x=\frac{\sqrt{21}+1}{5}$$

Steps for Completing the Square

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $x$.

$$x\times 2-5xx=0-4$$

Multiply $x$ and $x$ to get $x^{2}$.

$$x\times 2-5x^{2}=0-4$$

Subtract $4$ from $0$ to get $-4$.

$$x\times 2-5x^{2}=-4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-5x^{2}+2x=-4$$

Divide both sides by $-5$.

$$\frac{-5x^{2}+2x}{-5}=-\frac{4}{-5}$$

Dividing by $-5$ undoes the multiplication by $-5$.

$$x^{2}+\frac{2}{-5}x=-\frac{4}{-5}$$

Divide $2$ by $-5$.

$$x^{2}-\frac{2}{5}x=-\frac{4}{-5}$$

Divide $-4$ by $-5$.

$$x^{2}-\frac{2}{5}x=\frac{4}{5}$$

Divide $-\frac{2}{5}$, the coefficient of the $x$ term, by $2$ to get $-\frac{1}{5}$. Then add the square of $-\frac{1}{5}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{4}{5}+\left(-\frac{1}{5}\right)^{2}$$

Square $-\frac{1}{5}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{4}{5}+\frac{1}{25}$$

Add $\frac{4}{5}$ to $\frac{1}{25}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{21}{25}$$

Factor $x^{2}-\frac{2}{5}x+\frac{1}{25}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{21}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{21}{25}}$$

Simplify.

$$x-\frac{1}{5}=\frac{\sqrt{21}}{5}$$ $$x-\frac{1}{5}=-\frac{\sqrt{21}}{5}$$

Add $\frac{1}{5}$ to both sides of the equation.

$$x=\frac{\sqrt{21}+1}{5}$$ $$x=\frac{1-\sqrt{21}}{5}$$