Solving Quadratics from a Graph

Worked example

1. 1

   $x^2-6x+4=0$ — The equation. Its left side is the curve on your grid, so you want the points where y = 0.

2. 2

   $y=0\text{is the x-axis}$ — Every point on the x-axis has y = 0, so look where the curve crosses that axis.

3. 3

   $\text{left crossing: }x\approx 0.8$ — Read straight down from the crossing to the scale. It is a little before 1, so x ≈ 0.8 to 1 decimal place.

4. 4

   $\text{right crossing: }x\approx 5.2$ — The U crosses the axis a second time, just past 5, so x ≈ 5.2.

5. 5

   $\frac{0.8+5.2}{2}=3$ — Quick check: the two answers sit evenly either side of the line of symmetry x = 3. Their average is 3, so the readings are sensible.

6. 6

   $x\approx 0.8\text{or}x\approx 5.2$ — Write both x values. Do not write points like (0.8, 0): the question asks for x.

Worked example

1. 1

   $x^2-6x+4=3$ — The equation. The left side is the curve you already have; the right side is just the number 3.

2. 2

   $\text{draw }y=3$ — On the curve, y stands for $x^2-6x+4$. So “curve = 3” means y = 3: draw that flat line right across the grid.

3. 3

   $\text{left cut: }x\approx 0.2$ — Find where the line cuts the curve on the left, then read straight down to the x-axis: x ≈ 0.2.

4. 4

   $\text{right cut: }x\approx 5.8$ — Do the same on the right side of the U: x ≈ 5.8.

5. 5

   $\frac{0.2+5.8}{2}=3$ — Check: the average is 3, the line of symmetry again. Both readings agree.

Worked example

1. 1

   $\text{lowest point }(3,-5)$ — The bottom of the U. The curve never goes below y = −5, so −5 is the number to compare with.

2. 2

   $k>-5\Rightarrow \text{two solutions}$ — The flat line y = k is above the bottom, so it cuts both sides of the U. Example: k = 3 gave 0.2 and 5.8.

3. 3

   $k=-5\Rightarrow \text{one solution, }x=3$ — The line just touches the bottom of the U, at its one lowest point.

4. 4

   $k<-5\Rightarrow \text{no solutions}$ — The line is below the whole curve, so they never meet. Example: k = −6.

Worked example

1. 1

   $-x^2+2x+4=0$ — The equation. The minus on $x^2$ turns the U upside down, but you still look where y = 0: the x-axis.

2. 2

   $\text{highest point }(1,5)$ — The top of the ∩. The curve never goes above y = 5.

3. 3

   $\text{left crossing: }x\approx -1.2$ — Read down from where the curve crosses the axis on the left. It is past −1, so x ≈ −1.2, with its minus sign.

4. 4

   $\text{right crossing: }x\approx 3.2$ — The other crossing is just past 3: x ≈ 3.2.

5. 5

   $\frac{(-1.2)+3.2}{2}=1$ — Check: −1.2 + 3.2 = 2, and 2 ÷ 2 = 1, the line of symmetry x = 1. The readings agree.

6. 6

   $-x^2+2x+4=6\Rightarrow \text{no solutions}$ — For “how many”, flip the rule: 6 is above the highest point 5, so the line y = 6 misses the curve.