A. Quadratic Functions · Quadratic Expressions and Equations

Choosing a Strategy: Graphing, Factoring or the Quadratic Formula

A1.8About 70 minutes

Learning goals

I can solve a variety of quadratic equations by graphing, factoring, square roots or the quadratic formula.
I can choose an efficient strategy by looking at the equation.
I can describe the advantages and disadvantages of each strategy.

Words to know

strategy
A method chosen to solve a problem.

efficient
Getting a correct answer with the least work and fewest chances for error.

point of intersection
A point where two graphs meet; its x-coordinate solves the equation formed by setting the two expressions equal.

estimate
An approximate value, such as a root read from a graph.

Three tools, one job

You now have three main ways to find the real roots of a quadratic equation: graphing, factoring and the quadratic formula (plus square roots for equations shaped like (x + p)² = k). The curriculum asks you to solve a variety of equations, such as 100x² = 115x + 35, and to describe the advantages and disadvantages of each strategy. A skilled solver does not use the same tool every time: they look at the equation first and choose.

Strategy	Advantages	Disadvantages
Graphing (technology or by hand)	Shows the number of roots at a glance; gives a picture; works for any equation, including ones with two sides graphed separately; great for checking	By hand, gives only estimates; technology may round; you must still interpret which points matter
Factoring	Fast and exact when it works; shows the factors and x-intercepts directly	Only works when the expression factors over the integers; can be slow with large numbers
Quadratic formula	Always works; gives exact roots (with radicals) or decimals; the discriminant tells you how many roots first	More arithmetic and more chances for sign errors; slower than factoring for easy equations
Square roots	Very fast for (x + p)² = k or ax² = k	Only works when there is no separate x-term

Advantages and disadvantages of each solving strategy.

Worked example

Example 1: The curriculum's equation, three ways

Solve 100x² = 115x + 35 by factoring, by the formula and by graphing.

Rearrange first: 100x² − 115x − 35 = 0.
Factoring: 5(20x² − 23x − 7) = 5(5x − 7)(4x + 1) = 0, so x = 1.4 or x = −0.25. (Finding −28 and 5 for the decomposition took some searching: ac = −140.)
Formula: a = 100, b = −115, c = −35. D = 13 225 + 14 000 = 27 225 = 165². x = (115 ± 165) / 200, so x = 280/200 = 1.4 or x = −50/200 = −0.25.
Graphing: graph y = 100x² − 115x − 35 and read the x-intercepts: −0.25 and 1.4.
Comparison: all three agree. Factoring was quick once the GCF 5 was removed, but finding the split took trial; the formula needed big arithmetic but no guessing; graphing gave a picture and a check.

x = 1.4 or x = −0.25

Graphing technology shows the two x-intercepts of y = 100x² − 115x − 35 at x = −0.25 and x = 1.4, confirming Example 1.

A decision guide for choosing a strategy. Graphing fits with every path: use it to predict or check.

Worked example

Example 2: Choose, then solve

Choose a strategy for each equation, explain your choice, and solve. (a) x² − 9x + 20 = 0 (b) 2x² + 3x − 7 = 0 (c) (x − 4)² = 10 (d) 0.3x² − 1.7x + 0.9 = 0

(a) Factor: small integers, product 20 and sum −9 gives −4 and −5. (x − 4)(x − 5) = 0, so x = 4 or x = 5.
(b) D = 9 + 56 = 65, not a perfect square, so it will not factor: formula. x = (−3 ± √65)/4 ≈ 1.27 or −2.77.
(c) Already (something)² = k: square roots. x − 4 = ±√10, x = 4 ± √10 ≈ 7.16 or 0.84.
(d) Decimals: formula (or graphing technology). D = 2.89 − 1.08 = 1.81, √1.81 ≈ 1.3454. x = (1.7 ± 1.3454)/0.6 ≈ 5.08 or 0.59.

(a) 4, 5 (b) ≈ 1.27, −2.77 (c) ≈ 7.16, 0.84 (d) ≈ 5.08, 0.59

Graphing both sides

You do not always have to rearrange before graphing. To solve x² − 2x = 3, graph y = x² − 2x and y = 3 on the same axes. The solutions are the x-coordinates of the points where the graphs meet. This is handy with technology, and it shows clearly why there can be two, one or no solutions: a horizontal line can cut a parabola twice, touch it once or miss it.

y = x² − 2x and y = 3 meet at (−1, 3) and (3, 3), so the solutions of x² − 2x = 3 are x = −1 and x = 3. (Check: x² − 2x − 3 = (x − 3)(x + 1).)

A warning about graphs

Graphs are excellent for estimating and checking, but they can mislead when a parabola comes very close to the x-axis. The graph of y = x² − 6x + 9.01 seems to touch the axis at x = 3, but D = 36 − 36.04 = −0.04 < 0: it has no real roots. When it matters, confirm with algebra.

Just for fun

Using the quadratic formula on x² − 4 = 0 is like bringing a snowplough to clear one snowflake off the front step. It works, but the neighbours will wonder.

Hands-on activity

Method race

You need: a timer (phone), paper, a pencil, a calculator, and a family member or friend.

Write these on cards: x² − 5x + 6 = 0; 3x² + 2x − 4 = 0; (x + 3)² = 12; x² − 25 = 0; 2x² − x − 6 = 0; 0.4x² + x − 2 = 0.
Shuffle. For each card, first say which method you will use and why, then time yourself solving it.
Your partner solves the same card with the formula every time. Compare times and answers.
Write one sentence about when the formula was slower and when it was the best choice.

What to look for: answers: 2 and 3 (factor); about 0.87 and −1.54 (formula); about 0.46 and −6.46 (square roots: −3 ± √12); ±5 (difference of squares); 2 and −1.5 (factor: (x − 2)(2x + 3)); about 1.31 and −3.81 (formula). Factoring and square roots should win on the cards built for them; the formula wins (or ties) on the others.

Your turn

Choose a method and solve x² + 2x − 24 = 0.

Choose a method and solve 3x² − x − 5 = 0.

Choose a method and solve 2(x + 1)² = 18.

Choose a method and solve 4x² − 20x + 25 = 0.

Choose a method and solve x² + 4 = 2x.

Choose a method and solve 1.2x² + 0.5x − 2 = 0.

Give one advantage and one disadvantage of solving by graphing.

More practice: warm-up

Which method is fastest for (x − 2)² = 5? Solve it.

Which method is fastest for x² − 7x + 12 = 0? Solve it.

Which method always works?

Which method gives you a picture of the roots?

Name one disadvantage of factoring.

More practice: on your own

Solve 6x² = 11x + 10. Which method did you choose and why?

Solve x² + 1 = 3x + 5 by thinking about the graphs of both sides, then check algebraically.

Solve 2.5x² − 4x − 3 = 0.

Solve x(x − 8) = −16.

Kim used the quadratic formula to solve x² − 16 = 0. Suggest a faster way and solve.

Challenge

A stone is thrown from a 30 m cliff: h = −4.9t² + 12t + 30. When does it hit the water at the base of the cliff (h = 0)? Explain your method choice.

Solve 3x² − 12x + 7 = 0 by completing the square and by the formula. Which was easier for you?

A graph of y = x² − 6x + 9.01 drawn by hand seems to touch the x-axis at x = 3. Is x = 3 a root? How many real roots are there?

Check your understanding

Quiz

Which strategy is best for x² − 6x + 8 = 0?

Which strategy is best for (x + 5)² = 7?

One disadvantage of the quadratic formula is:

To solve x² + 2 = 4x by graphing both sides, you look for:

The roots of 100x² = 115x + 35 are:

Why should you check a hand-drawn graph with algebra when the vertex is very close to the x-axis?

For parents and teachers

Big idea (A1.8): students determine real roots of a variety of quadratic equations and describe the advantages and disadvantages of each strategy (graphing, factoring, the quadratic formula).

How to help: before your teen starts any equation, ask, "Which method, and why?" There is often more than one good answer; the reasoning is what matters.

Common difficulties:

Using the formula for everything, even x² − 16 = 0, and making avoidable arithmetic errors.
Trying to factor equations with decimal coefficients.
Forgetting to rearrange to ax² + bx + c = 0 before factoring or using the formula.
Reading y-coordinates instead of x-coordinates when graphing both sides.