use the quadratic formula to solve this 4x^2+4x=7

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.

Use the quadratic formula to solve this 4x^2 + 4x = 7

Learn how to solve the quadratic equation 4x² + 4x = 7 step by step using the quadratic formula. This detailed solution covers key algebraic concepts such as discriminants and square roots, making it ideal for students in Grades 9-11.

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