Let’s solve each quadratic equation by graphing — that means we’ll find where the parabola crosses the x-axis (those are the solutions, also called roots or zeros).
We’ll go one problem at a time.
---
Problem 1: -x² + 2x + 3 = 0
Step 1: Rewrite in standard form if needed → already is:
f(x) = -x² + 2x + 3
Step 2: Find key points to graph:
- Vertex: x = -b/(2a) = -2 / (2 * -1) = -2 / -2 = 1
Plug x=1 into equation: f(1) = -(1)² + 2(1) + 3 = -1 + 2 + 3 = 4 → vertex at (1, 4)
- Y-intercept: set x=0 → f(0) = 0 + 0 + 3 = 3 → (0, 3)
- X-intercepts: solve -x² + 2x + 3 = 0 → multiply both sides by -1: x² - 2x - 3 = 0 → factor: (x - 3)(x + 1) = 0 → x = 3 or x = -1
So the graph crosses x-axis at x = -1 and x = 3.
✔ Solution(s):
x = -1, 3
---
Problem 2: 3x² = 6x - 3
Step 1: Move all terms to one side:
3x² - 6x + 3 = 0
Step 2: Simplify by dividing by 3:
x² - 2x + 1 = 0
Step 3: Factor: (x - 1)² = 0 → x = 1 (double root)
Graph touches x-axis at only one point: x = 1
✔ Solution(s):
x = 1
---
Problem 3: -8 = -x² - 4
Step 1: Add 8 to both sides:
0 = -x² + 4 → or rewrite as: -x² + 4 = 0
Multiply both sides by -1: x² - 4 = 0 → (x - 2)(x + 2) = 0 → x = 2 or x = -2
Check original: plug x=2 → -8 = -(4) - 4 → -8 = -8 ✔️
x=-2 → same since squared → -8 = -4 -4 → -8 = -8 ✔️
✔ Solution(s):
x = -2, 2
---
Problem 4: 3x = (1/4)x² + 5
Step 1: Move all terms to one side:
0 = (1/4)x² - 3x + 5
Multiply entire equation by 4 to eliminate fraction:
0 = x² - 12x + 20
Factor: look for two numbers that multiply to 20 and add to -12 → -10 and -2
→ (x - 10)(x - 2) = 0 → x = 10 or x = 2
Check in original:
For x=2: Left: 3*2=6; Right: (1/4)*4 + 5 = 1 + 5 = 6 ✔️
For x=10: Left: 30; Right: (1/4)*100 + 5 = 25 + 5 = 30 ✔️
✔ Solution(s):
x = 2, 10
---
Final Answer:
1. x = -1, 3
2. x = 1
3. x = -2, 2
4. x = 2, 10
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations by graphing worksheet.
