Question

Solve the equation by using the quadratic formula.
3x² - 1 = 7x

Answer 1

Answer:

• Solutions:   $\displaystyle\mathsf{x\:=\:\frac{7+ \sqrt{61}}{6}}$ ,   $\displaystyle\mathsf{x\:=\:\frac{7 - \sqrt{61}}{6}}$

Step-by-step explanation:

We are given the following quadratic equation, 3x² - 1 = 7x, which requires us to find its possible solutions using the quadratic formula.  

Definitions:

• In a quadratic equation, the leading term has a second-degree (or squared) variable.  
• The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are real numbers, and that a ≠ 0.

Quadratic Formula:

The quadratic formula is given by:

$\displaystyle\mathsf{x\:=\:\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

We will use this formula to solve for the roots or the solution of the given quadratic equation.

Solution:

Step 1: transform the given equaton into standard form.

First, we must subtract "7x" from both sides of the equation:

3x² - 7x - 1 = 7x - 7x

3x² - 7x - 1 = 0     ⇒   This is the standard form of the given quadratic equation where a = 3, b = -7, and c = - 1.

Step 2: Plug in the values for a, b, & c into the quadratic formula.

The derived values for a, b, & c from the previous step are:  a = 3, b = -7, and c = - 1.  Substitute these values into the following quadratic formula:

$\displaystyle\mathsf{x\:=\:\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

$\displaystyle\mathsf{x\:=\:\frac{-(-7)\pm\sqrt{(-7)^2-4(3)(-1)}}{2(3)}}$

Step 3: Following the PEMDAS Order of Operations, start by applying the Power rule of Exponents  $\displaystyle\mathsf{(a^m)^n = (a^{m*n})}$ (under the radical) :

$\displaystyle\mathsf{x\:=\:\frac{-(-7)\pm\sqrt{49-4(3)(-1)}}{2(3)}}$

Step 4: Next, multiply the integers under the radical, PEMDAS Order of Operations.

$\displaystyle\mathsf{x\:=\:\frac{-(-7)\pm\sqrt{61}}{2(3)}}$

Step 5: Multiply the integers in the denominator, and apply the multiplication integer rule, -(-a) = a.

$\displaystyle\mathsf{x\:=\:\frac{7\pm\sqrt{61}}{6}}$

•  Since $\displaystyle\mathsf{\sqrt{61}}$ is not a perfect square, it can remain the same.

Step 6: Separate the solutions.

$\displaystyle\mathsf{x\:=\:\frac{7+ \sqrt{61}}{6}}$  , and   $\displaystyle\mathsf{x\:=\:\frac{7 - \sqrt{61}}{6}}$  

Final Answer:

Therefore, the solution to 3x² - 1 = 7x  are:  $\displaystyle\mathsf{x\:=\:\frac{7+ \sqrt{61}}{6}}$,  $\displaystyle\mathsf{x\:=\:\frac{7 - \sqrt{61}}{6}}$.

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Keywords:

Quadratic equation

Quadratic function

Standard form

Parabola

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