To solve the quadratic equation $4x^2 + 4x = 7$ using the quadratic formula, let's follow these steps:

Step 1: Rearrange the equation

First, rewrite the equation in standard quadratic form: $4x^2 + 4x - 7 = 0$ Now we can see it's a quadratic equation of the form: $ax^2 + bx + c = 0$ where:

• $a = 4$
• $b = 4$
• $c = -7$

Step 2: Apply the quadratic formula

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substitute $a = 4$, $b = 4$, and $c = -7$ into the formula:

$x = \frac{-4 \pm \sqrt{4^2 - 4(4)(-7)}}{2(4)}$

Step 3: Simplify

First, calculate the discriminant $b^2 - 4ac$:

$b^2 - 4ac = 4^2 - 4(4)(-7) = 16 + 112 = 128$

Now substitute this into the quadratic formula:

$x = \frac{-4 \pm \sqrt{128}}{8}$

Step 4: Simplify further

The square root of 128 can be simplified:

$\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$

So the equation becomes:

$x = \frac{-4 \pm 8\sqrt{2}}{8}$

Step 5: Split into two solutions

Now, divide both terms by 8:

$x = \frac{-4}{8} \pm \frac{8\sqrt{2}}{8}$ $x = -\frac{1}{2} \pm \sqrt{2}$

Final Answer:

The two solutions are: $x = -\frac{1}{2} + \sqrt{2} \quad \text{or} \quad x = -\frac{1}{2} - \sqrt{2}$

Would you like more details on any step, or have any questions?

Related Questions:

1. How do you find the discriminant of a quadratic equation?
2. Can you explain how to simplify square roots?
3. What are the properties of quadratic functions?
4. How does the quadratic formula work for complex roots?
5. How do you factor a quadratic equation?

Tip:

Always check the discriminant ($b^2 - 4ac$) before using the quadratic formula. It tells you how many real solutions to expect.