Grade 10 Quadratic Equations

Quadratic Equation: A polynomial equation of the second degree in one variable.

Standard Form: A quadratic equation can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers and 'a' is not equal to zero (a $\neq$ 0).

Example: 2x² + 3x - 5 = 0 is a quadratic equation (a=2, b=3, c=-5).
x² - 4x = 0 can be written as x² - 4x + 0 = 0 (a=1, b=-4, c=0).
3x² - 6 = 0 can be written as 3x² + 0x - 6 = 0 (a=3, b=0, c=-6).

Non-Examples:
x³ + x² + 1 = 0 (not quadratic, highest power is 3)
x + 5 = 0 (not quadratic, highest power is 1)

Roots: The values of the variable (usually x) that satisfy the quadratic equation are called its roots or solutions.

Number of Roots: A quadratic equation has at most two roots. These roots can be real or complex. In Grade 10, we primarily focus on real roots.

Example: Consider the equation x² - 4 = 0.

• If x = 2, then 2² - 4 = 4 - 4 = 0. This is true. So, x = 2 is a root.
• If x = -2, then (-2)² - 4 = 4 - 4 = 0. This is true. So, x = -2 is a root.
• If x = 3, then 3² - 4 = 9 - 4 = 5, which is not equal to 0. So, x = 3 is NOT a root.

Geometric Interpretation: The real roots of a quadratic equation ax² + bx + c = 0 are the x-intercepts of the graph of the corresponding quadratic function y = ax² + bx + c (which is a parabola).

Method: Factorize the quadratic expression ax² + bx + c into the product of two linear factors. Then, set each factor equal to zero and solve for x. This method relies on the property that if the product of two numbers is zero, at least one of them must be zero.

Example: Solve x² - 5x + 6 = 0 by factorization.

Step 1: Factorize the expression x² - 5x + 6. We look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.
x² - 5x + 6 = (x - 2)(x - 3)

Step 2: Set each factor equal to zero.
(x - 2) = 0 OR (x - 3) = 0

Step 3: Solve each linear equation for x.
x - 2 = 0 => x = 2
x - 3 = 0 => x = 3

Result: The roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Quadratic Formula: For a quadratic equation in the standard form ax² + bx + c = 0 (where a $\neq$ 0), the roots are given by the formula:
x = [-b $\pm$ $\sqrt{b^2 - 4ac}$] / 2a

Discriminant ($\Delta$): The expression inside the square root, b² - 4ac, is called the discriminant. It determines the nature of the roots.

Example: Solve 2x² + 5x - 3 = 0 using the quadratic formula.
Here, a = 2, b = 5, c = -3.

Step 1: Calculate the discriminant ($\Delta$).
$\Delta$ = b² - 4ac = 5² - 4(2)(-3) = 25 - (-24) = 25 + 24 = 49.

Step 2: Use the quadratic formula.
x = [-b $\pm$ $\sqrt{\Delta}$] / 2a
x = [-5 $\pm$ $\sqrt{49}$] / 2(2)
x = [-5 $\pm$ 7] / 4

Step 3: Find the two roots using the $\pm$ sign.
Root 1: x = (-5 + 7) / 4 = 2 / 4 = 1/2
Root 2: x = (-5 - 7) / 4 = -12 / 4 = -3

Result: The roots of the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.

Discriminant ($\Delta$): $\Delta$ = b² - 4ac. The value of the discriminant tells us about the nature of the roots of the quadratic equation ax² + bx + c = 0.

Case 1: $\Delta$ > 0 (Discriminant is positive)
The equation has two distinct real roots. The parabola intersects the x-axis at two different points.

Case 2: $\Delta$ = 0 (Discriminant is zero)
The equation has two equal real roots (or one real root with multiplicity 2). The parabola touches the x-axis at exactly one point.

Case 3: $\Delta$ < 0 (Discriminant is negative)
The equation has no real roots. It has two complex (non-real) roots. The parabola does not intersect the x-axis.