Formation of Quadratic Equations with Given Roots – SS1 Mathematics Lesson Note (First Term)

A quadratic equation with given roots is formed by writing x² – (sum of roots)x + (product of roots) = 0.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state the relationship between the roots of a quadratic equation and its coefficients;
2. form a quadratic equation from two given roots using the factor method;
3. form a quadratic equation from two given roots using the sum and product of the roots;
4. form quadratic equations whose roots are fractions or negative numbers, with whole-number coefficients;
5. draw the graph of a quadratic expression and read the roots from the points where the curve meets the x-axis;
6. state the maximum or minimum value of y and the line of symmetry from the graph.

What is a quadratic equation and what are its roots?

A quadratic equation is an equation whose highest power of the unknown is 2, written as ax² + bx + c = 0, where a is not zero.

• Root: a value of x that makes the equation true. Roots are also called solutions or zeros.
• Number of roots: a quadratic equation has at most two roots.
• Equal roots: the two roots can be the same number, as in x = 3 twice.
• Standard form: ax² + bx + c = 0, with the right-hand side equal to zero.

What are the sum and product of the roots?

If the roots of a quadratic equation are α and β, the sum of the roots is α + β and the product is αβ.

• Sum of roots: α + β, added as they are given.
• Product of roots: α × β, multiplied as they are given.
• Equation when a = 1: x² – (α + β)x + αβ = 0.
• Link to coefficients: for ax² + bx + c = 0, the sum of roots is -b/a and the product is c/a.

How do you form a quadratic equation from given roots?

There are two methods: the factor method and the sum and product method.

Method 1: The factor method

If x = α, then (x – α) is a factor. Multiply the two factors and equate to zero.

1. Write each root as x = root.
2. Move the number to the left side to make each side zero: x – root = 0.
3. Write the product of the two brackets: (x – α)(x – β) = 0.
4. Expand the brackets and collect like terms.
5. Write the answer in the form ax² + bx + c = 0.

Method 2: The sum and product method

1. Find the sum of the roots, α + β.
2. Find the product of the roots, αβ.
3. Substitute into x² – (sum)x + product = 0.
4. If there are fractions, multiply every term by the common denominator.

What are some worked examples of forming quadratic equations?

Example A: Roots 2 and 3

• Factor method: (x – 2)(x – 3) = 0, so x² – 3x – 2x + 6 = 0.
• Answer: x² – 5x + 6 = 0.
• Check: sum = 2 + 3 = 5 and product = 2 × 3 = 6.

Example B: Roots -4 and 1

• Sum of roots: -4 + 1 = -3.
• Product of roots: (-4)(1) = -4.
• Substitute: x² – (-3)x + (-4) = 0.
• Answer: x² + 3x – 4 = 0.

Example C: Roots 1/2 and -3

• Write the factors: x = 1/2 gives 2x – 1 = 0, and x = -3 gives x + 3 = 0.
• Multiply: (2x – 1)(x + 3) = 2x² + 6x – x – 3.
• Answer: 2x² + 5x – 3 = 0.
• Check by sum: 1/2 + (-3) = -5/2, which equals -b/a = -5/2.

Example D: Roots -2/3 and 4

• Write the factors: x = -2/3 gives 3x + 2 = 0, and x = 4 gives x – 4 = 0.
• Multiply: (3x + 2)(x – 4) = 3x² – 12x + 2x – 8.
• Answer: 3x² – 10x – 8 = 0.

Example E: Equal roots, 3 and 3

• Factors: (x – 3)(x – 3) = 0.
• Answer: x² – 6x + 9 = 0.

Summary table of roots and their equations

What mistakes should you avoid?

• Sign errors: a root of -4 gives the factor (x + 4), not (x – 4).
• Wrong sign on the sum: the middle term is minus the sum of the roots.
• Leaving fractions: clear them by multiplying through by the common denominator.
• Missing “= 0”: an equation must end with “= 0”.

How do you solve a quadratic equation by the graphical method?

The graphical method finds the roots of a quadratic equation from the points where the curve of its expression meets the x-axis.

1. Use the given range of values of the independent variable (usually x) to find the corresponding values of the dependent variable (usually y) from the quadratic equation or relation given. If the range of values is not given, choose a suitable one.
2. From the results in step 1, prepare a table of values for the given quadratic expression.
3. Choose a suitable scale for the graph.
4. Draw the axes and plot the points.
5. Join the points to form a smooth curve, drawing freehand or using a flexible curve.

What should you know about the graph of a quadratic equation?

• Roots: the roots are the x values where the curve cuts the x-axis, because y = 0 along the x-axis.
• Inverted n-shape: the curve is an inverted n-shaped parabola when the coefficient of x² is negative.
• V-shape: the curve is a V-shaped parabola when the coefficient of x² is positive.
• Maximum value: the maximum value of y is at the highest point of the n-shaped parabola.
• Minimum value: the minimum value of y is at the lowest point of the V-shaped parabola.
• Line of symmetry: the line which divides the curve into two equal parts.

What are the three positions of the curve to the x-axis?

• Cuts at two points: the equation has two different roots.
• Touches at one point: the equation has equal roots (one repeated root).
• Does not meet the axis: the equation has no real roots.

Worked examples on the graphical method

Example 1

1. Draw the graph of y = 11 + 8x – 2x² from x = -2 to x = +6.
2. Hence find the approximate roots of the equation 2x² – 8x – 11 = 0.
3. From the graph, find the maximum value of y.

Solution to (a)

y = 11 + 8x – 2x², from x = -2 to x = +6.

• When x = -2: y = 11 + 8(-2) – 2(-2)² = 11 – 16 – 2(4) = 11 – 16 – 8 = -13
• When x = -1: y = 11 + 8(-1) – 2(-1)² = 11 – 8 – 2(1) = 11 – 8 – 2 = 1
• When x = 0: y = 11 + 8(0) – 2(0)² = 11 + 0 – 0 = 11
• When x = 1: y = 11 + 8(1) – 2(1)² = 11 + 8 – 2 = 17
• When x = 2: y = 11 + 8(2) – 2(2)² = 11 + 16 – 8 = 19
• When x = 3: y = 11 + 8(3) – 2(3)² = 11 + 24 – 18 = 17
• When x = 4: y = 11 + 8(4) – 2(4)² = 11 + 32 – 32 = 11
• When x = 5: y = 11 + 8(5) – 2(5)² = 11 + 40 – 50 = 1
• When x = 6: y = 11 + 8(6) – 2(6)² = 11 + 48 – 72 = -13

Solution to (b)

The approximate roots are the x values where the curve cuts the x-axis. This is so because:

1. y = 11 + 8x – 2x²
2. Multiply both sides by -1: -y = -11 – 8x + 2x²
3. So -y = 2x² – 8x – 11
4. When 2x² – 8x – 11 = 0, then -y = 0, i.e. y = 0.

Thus, from the graph, the roots of 2x² – 8x – 11 = 0 are x = -1.1 or x = 5.1.

Solution to (c)

The maximum value of y = 19.

Example 2

Given that y = 4x² – 12x + 9, (a) copy and complete the table below.

2. Hence draw a graph and find the roots of the equation 4x² – 12x + 9 = 0.
3. From the graph, what is the minimum value of y?
4. From the graph, what is the line of symmetry of the curve?

Solution to (a)

Solution to (b), (c) and (d)

• The curve touches the x-axis, so the roots are x = 1.5 twice.
• The minimum value of y = 0.
• The line of symmetry of the curve is the line x = 1.5.

Evaluation

1. Form the quadratic equation whose roots are 3 and -5.
2. Form the quadratic equation, with whole-number coefficients, whose roots are 1/2 and 4.
3. State the sum and the product of the roots of x² – 7x + 10 = 0.
4. State the steps for solving a quadratic equation by the graphical method.
5. Why are the roots of a quadratic equation found where the curve meets the x-axis?

Assignment

1. Form the quadratic equation whose roots are -4 and 1.
2. Prepare a table of values for the graph of y = x² + 3x – 4 for values of x from -6 to +3.
3. Use a scale of 1 cm to 1 unit on both axes and draw the graph.
4. Find the least value of y.
5. Read the roots of x² + 3x – 4 = 0 from the graph and compare them with the roots in (a).
6. Find the values of x when y = 1.