Solve quadratic equations by using different methods (factorisation, completing the square, and quadratic formula)

Not all quadratic equations can be solved by factorization. Completing the square is a method that works for all quadratic equations. The idea is to transform the equation so that the left side becomes a perfect square.

Key Idea

For an expression like $x^2 + bx$, we add $(\frac{b}{2})^2$ to make it a perfect square: $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Steps for Completing the Square

1. Write the equation in the form $ax^2 + bx + c = 0$
2. If $a \neq 1$, divide both sides by $a$
3. Move the constant term to the right side
4. Add $(\frac{b}{2})^2$ to both sides
5. Write the left side as a perfect square
6. Take the square root of both sides
7. Solve for $x$

Worked Example

Solve $x^2 + 6x - 7 = 0$ by completing the square

Solution

Step 1: The equation is already in standard form with $a = 1$.

Step 2: Divide by 1 (no change needed).

Step 3: Move constant to the right side: $x^2 + 6x = 7$

Step 4: Add $(\frac{6}{2})^2 = 9$ to both sides: $x^2 + 6x + 9 = 7 + 9$ $x^2 + 6x + 9 = 16$

Step 5: Write the left side as a perfect square: $(x + 3)^2 = 16$

Step 6: Take the square root of both sides: $x + 3 = \pm 4$

Step 7: Solve:

• If $x + 3 = 4$, then $x = 1$
• If $x + 3 = -4$, then $x = -7$

Therefore, $x = 1$ or $x = -7$

The quadratic formula is a universal method that can solve any quadratic equation. It is derived from completing the square and works even when factorization is difficult or impossible.

The Quadratic Formula

For any quadratic equation $ax^2 + bx + c = 0$ (where $a \neq 0$): $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root, $b^2 - 4ac$, is called the discriminant. It tells us about the nature of the roots:

• If $b^2 - 4ac > 0$, there are two distinct real roots
• If $b^2 - 4ac = 0$, there is one repeated real root
• If $b^2 - 4ac < 0$, there are no real roots

Steps for Using the Formula

1. Identify $a$, $b$, and $c$ from the equation
2. Calculate the discriminant $b^2 - 4ac$
3. Substitute into the formula
4. Simplify to find both solutions

Worked Example

Solve $3x^2 + 5x - 2 = 0$ using the quadratic formula

Solution

Step 1: Identify $a = 3$, $b = 5$, $c = -2$

Step 2: Calculate the discriminant: $b^2 - 4ac = 5^2 - 4(3)(-2) = 25 + 24 = 49$

Step 3: Substitute into the formula: $x = \frac{-5 \pm \sqrt{49}}{2(3)} = \frac{-5 \pm 7}{6}$

Step 4: Find both solutions:

• $x = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3}$
• $x = \frac{-5 - 7}{6} = \frac{-12}{6} = -2$

Therefore, $x = \frac{1}{3}$ or $x = -2$

• Factorization: Use when the quadratic can be easily factored into whole numbers
• Completing the square: Use when factorization is difficult, or when learning the technique for future topics
• Quadratic formula: Use when other methods are difficult or as a reliable fallback

Quadratic equations are used in many real-life situations in Tanzania. For example, a shopkeeper at a market in Dar es Salaam may use quadratic equations to calculate pricing. If a trader wants to set the price of mangoes so that the revenue meets a target, the problem can be written as a quadratic equation and solved to find the correct selling price. Similarly, farmers can use these equations to calculate crop yields or determine dimensions for storage containers, helping them plan for harvests and sales.