Quadratic equations

Factor out x: ax² + bx = x(ax + b) = 0. A product is zero when at least one factor is zero. Thus x = 0 or ax + b = 0, giving x = −b/a.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.

The discriminant is an expression formed from the coefficients of a quadratic equation ax² + bx + c = 0. It helps determine the number of real solutions.

Formula: D = b² − 4ac.

1. Identify the coefficients a, b and c, paying attention to their signs.
2. Calculate b².
3. Calculate 4ac and subtract it from b².

If D > 0, the equation has two real solutions; if D = 0, it has one; if D < 0, it has no real solutions.

An equation root is a value of x that makes the equation true when substituted for x.

For this question, you do not need to solve the whole equation. Substitute the given number for every x and calculate the left-hand side:

x² + bx + c.

1. Square the given number.
2. Multiply the given number by the coefficient b.
3. Add c and combine all the resulting numbers.

If the result is 0, the given number is a root of the equation. If the result is not 0, it is not a root.

When the roots of a quadratic equation must be found, first calculate the discriminant D = b² − 4ac, then use x = (−b ± √D) / 2a.

Let width be x and length x + d. The area gives x(x + d) = S, or x² + dx − S = 0. Solve and keep the positive width; length is x + d. Check their product.

For c > 0, x² = c has two roots: x = −√c and x = √c. Include the negative root because a negative number also has a positive square.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.