The Question 📝

Solve the quadratic equation $2x^2 + 6x - 7 = 0$ by completing the square.

Give your answer in exact surd form.

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The Solution

Here is the step-by-step method to solve the equation.

Step 1: Rearrange the Equation

First, move the constant term (the number without an $x$) to the right-hand side of the equation.

Step 2: Make the Coefficient of $x^2$ Equal to 1

The method of completing the square requires the coefficient of the $x^2$ term to be 1. To achieve this, divide every term in the equation by the current coefficient of $x^2$, which is 2.

This simplifies to:

Step 3: Complete the Square

Now, we need to find the value that "completes the square" on the left-hand side.

1. Take the coefficient of the $x$ term, which is 3.
2. Halve it: $\frac{3}{2}$.
3. Square the result: $(\frac{3}{2})^2 = \frac{9}{4}$.

Add this value to both sides of the equation to keep it balanced.

Step 4: Factorise and Simplify

The left-hand side is now a perfect square trinomial and can be factorised. The right-hand side can be simplified by finding a common denominator.

• Left-hand side: $x^2 + 3x + \frac{9}{4}$ factorises to $(x + \frac{3}{2})^2$.
• Right-hand side: To add the fractions, change $\frac{7}{2}$ to $\frac{14}{4}$. So, $\frac{14}{4} + \frac{9}{4} = \frac{23}{4}$.

Now the equation looks like this:

Step 5: Solve for $x$

To find $x$, we need to undo the operations.

1. Take the square root of both sides. Remember to include both the positive and negative roots ($\pm$).
   $$x + \frac{3}{2} = \pm\sqrt{\frac{23}{4}}$$
2. Simplify the square root on the right-hand side.
   $$x + \frac{3}{2} = \pm\frac{\sqrt{23}}{\sqrt{4}} = \pm\frac{\sqrt{23}}{2}$$
3. Isolate $x$ by subtracting $\frac{3}{2}$ from both sides.
   $$x = -\frac{3}{2} \pm \frac{\sqrt{23}}{2}$$

This gives us the two exact solutions: