4x2 + 10x − 3 = 0 by completing the square

To solve the quadratic equation \( 4x^2 + 10x - 3 = 0 \) by completing the square, follow these steps:

---

### Step 1: Divide through by the coefficient of \( x^2 \)
The coefficient of \( x^2 \) is 4, so divide the entire equation by 4 to simplify:
\[
x^2 + \frac{10}{4}x - \frac{3}{4} = 0
\]
Simplify further:
\[
x^2 + \frac{5}{2}x - \frac{3}{4} = 0
\]

---

### Step 2: Rearrange the equation
Move the constant term to the other side:
\[
x^2 + \frac{5}{2}x = \frac{3}{4}
\]

---

### Step 3: Complete the square
To complete the square, take half the coefficient of \( x \), square it, and add it to both sides. The coefficient of \( x \) is \( \frac{5}{2} \), so:
\[
\left( \frac{\frac{5}{2}}{2} \right)^2 = \left( \frac{5}{4} \right)^2 = \frac{25}{16}
\]
Add \( \frac{25}{16} \) to both sides:
\[
x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{3}{4} + \frac{25}{16}
\]

---

### Step 4: Simplify both sides
The left-hand side becomes a perfect square trinomial:
\[
\left( x + \frac{5}{4} \right)^2
\]
On the right-hand side, find a common denominator to combine the fractions:
\[
\frac{3}{4} = \frac{12}{16}, \quad \frac{3}{4} + \frac{25}{16} = \frac{12}{16} + \frac{25}{16} = \frac{37}{16}
\]
Thus:
\[
\left( x + \frac{5}{4} \right)^2 = \frac{37}{16}
\]

---

### Step 5: Solve for \( x \)
Take the square root of both sides:
\[
x + \frac{5}{4} = \pm \sqrt{\frac{37}{16}}
\]
Simplify:
\[
x + \frac{5}{4} = \pm \frac{\sqrt{37}}{4}
\]

Subtract \( \frac{5}{4} \) from both sides:
\[
x = -\frac{5}{4} \pm \frac{\sqrt{37}}{4}
\]

---

### Step 6: Combine terms
Combine into a single fraction:
\[
x = \frac{-5 \pm \sqrt{37}}{4}
\]

---

### Final Answer:
\[
x = \frac{-5 + \sqrt{37}}{4} \quad \text{or} \quad x = \frac{-5 - \sqrt{37}}{4}
\]

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Would you like a breakdown of any specific step?  
Here are some related questions you might want to explore:

1. How is completing the square related to the quadratic formula?
2. What are the applications of completing the square in real-world problems?
3. Why is \( \frac{-5 \pm \sqrt{37}}{4} \) in simplest form?
4. Can this method be used if the quadratic equation is already in vertex form?
5. How would the graph of this quadratic equation look?

**Tip:** Always ensure the coefficient of \( x^2 \) is 1 before completing the square—it simplifies the process significantly!

4x^2 + 10x − 3 = 0 by completing the square

Learn how to solve the quadratic equation 4x^2 + 10x − 3 = 0 step-by-step by completing the square. This detailed explanation covers algebraic concepts like rearranging equations, perfect square trinomials, and the square root principle. Suitable for Grades 8-10.

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