Solve quadratics graphically and numerically

How many x-axis crossings can you see?

Move the probe near each crossing of y=x²+x−3. Watch the function value change sign. A value close to zero helps locate a root; it is not automatically an exact zero

What should a numerical search aim for?

A root is a horizontal coordinate

To solve f(x)=0 graphically, find where the curve meets y=0 and read the x-values. Crossing twice gives two roots; touching once gives one distinct root; no meeting gives no real roots

Refine a graph reading numerically

A picture gives an estimate. Evaluate the rule near a crossing and narrow the interval where its sign changes. For a continuous quadratic, opposite signs at two inputs guarantee at least one root between them

Rule

\(\displaystyle f(x)=x^2+x-3\)

For f(x)=x²+x−3, calculate f(1)

Substitute 1 into the given rule

For f(x)=x²+x−3, given f(1.30)=−0.0100 and f(1.31)=0.0261, which interval contains a crossing?

Root bracket

\(\displaystyle 1.30<x<1.31\)

Round that positive root to one decimal place

Use the bracket to round the x-value, and write exactly one digit after the decimal point

Root bracket

\(\displaystyle -2.31<x<-2.30\)

A second search gives −2.31<x<−2.30. Round this root to one decimal place

Keep the negative sign and write exactly one digit after the decimal point

Use an approximation sign

The exact root is not 1.3 because f(1.3)=−0.01, not zero. Write x≈1.3 at one decimal place, and keep more digits for intermediate calculations

What does x≈1.3 mean here?

The graph shows y=f(x). How many distinct real solutions does f(x)=0 have?

The graph shows y=f(x). How many real solutions does f(x)=0 have?

Solve for a nonzero output level

To solve f(x)=L, find intersections of the curve with the horizontal line y=L. Alternatively, graph f(x)−L and find its x-intercepts

The dashed line is y=−2. How many distinct solutions does (x−1)²−2=−2 have?

Solve (x−1)²−2=2. Give the smaller solution

First isolate the squared term

Give the larger solution of the displayed equation

First isolate the squared term

Compare algebra with the graph

Expanding and collecting zero form gives x²−2x−3=0. Factorisation gives (x+1)(x−3)=0. The exact roots −1 and 3 agree with the two intersections at output 2.

Use the method independently

Use a graphing tool and a numerical table to solve x²−x−4=0 to two decimal places. Record a bracket for each root, test the original expression at the bounds, and explain why the rounded values are justified. Compare with the curve, label the precision and ask a teacher or peer to review your method. Answer check: the roots are approximately −1.56 and 2.56. The expression changes from positive to negative between −1.562 and −1.561, and from negative to positive between 2.561 and 2.562