Let’s solve each equation using the graph of y = x².
The idea is:
We want to find where the graph of y = x² meets another line (like y = 4, or y = x + 1, etc.), because that’s where the two sides of the equation are equal.
---
a) x² – 4 = 0
This means: x² = 4
So we look on the graph for points where y = x² and y = 4 meet.
On the graph, when y = 4, what x values give us that?
Look at the curve — it hits y=4 at x = -2 and x = 2.
✔ So solutions: x = -2 and x = 2
---
b) x² – x = 0
Rewrite: x² = x
Now we’re looking for where the parabola y = x² meets the line y = x.
Draw or imagine the line y = x (it goes through origin, slope 1).
Where do they cross?
At x = 0 → y = 0² = 0, and y = x = 0 → matches.
At x = 1 → y = 1² = 1, and y = x = 1 → also matches.
✔ So solutions: x = 0 and x = 1
---
c) x² – x – 1 = 0
Rewrite: x² = x + 1
Now we need to find where y = x² meets y = x + 1.
The line y = x + 1 has slope 1 and crosses y-axis at 1.
Look at the graph — does it intersect? Let’s estimate.
Try x = 1.5 → x² = 2.25; x+1 = 2.5 → too high
x = 1.6 → x² = 2.56; x+1 = 2.6 → close
x = 1.618... (golden ratio!) actually works exactly, but since we’re using graph, approximate.
From graph, you can see intersections near:
x ≈ -0.6 and x ≈ 1.6
(You can check: (-0.6)² = 0.36; -0.6 + 1 = 0.4 → close enough for graph)
✔ Approximate solutions: x ≈ -0.6 and x ≈ 1.6
*(Note: Exact values are [1±√5]/2, but since we’re using graph, approximation is fine.)*
---
d) 2x² – 3 = 0
Rewrite: 2x² = 3 → x² = 3/2 = 1.5
So now we look for where y = x² equals 1.5
On the graph, go up to y = 1.5, then move left/right to hit the curve.
That happens at about x = -1.2 and x = 1.2
(Check: 1.2² = 1.44 ≈ 1.5 — good enough for graph)
✔ Solutions: x ≈ -1.2 and x ≈ 1.2
---
Final Answer:
a) x = -2, 2
b) x = 0, 1
c) x ≈ -0.6, 1.6
d) x ≈ -1.2, 1.2
Parent Tip: Review the logic above to help your child master the concept of solve quadratic equations by graphing worksheet.
