SAT vs ACT – Difference between ACT and SAT

A quadratic equation can be solved by factoring if it can be expressed as the product of two binomials. Let's examine each option:

• a) x² + 5x + 6 = 0: This factors to (x + 2)(x + 3) = 0.
• b) x² + 3x - 4 = 0: This factors to (x + 4)(x - 1) = 0.
• c) x² + 6x + 8 = 0: This factors to (x + 2)(x + 4) = 0.
• d) 3x² + 2x + 1 = 0: This equation does not factor easily using integers. Its discriminant (b² - 4ac = 2² - 4 * 3 * 1 = 4 - 12 = -8) is negative, indicating complex roots, and is not factorable using real numbers.

Therefore, options a, b, and c can be solved by factoring, while option d is best solved using the quadratic formula or completing the square.

We are looking for two numbers that multiply to 8 and add up to 6. Those numbers are 2 and 4. Therefore, the factored form is (x + 2)(x + 4) = 0.

The equation x² + 8x + 16 = 0 can be factored as (x + 4)² = 0. Taking the square root of both sides gives x + 4 = 0, so x = -4. Because the factor (x+4) is squared, we say there is one real root with multiplicity 2, or a repeated root.

∛a² is equivalent to a raised to the power of 2/3.

A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent.

The power rule states that when raising a power to another power, you multiply the exponents.

While completing the square and graphing can also be used to solve quadratic equations, the quadratic formula is the most direct and generally applicable method, especially when factoring is difficult or not possible with integer or simple fractional roots. It provides a reliable way to find the solutions regardless of the complexity of the coefficients.

In vertex form y = a(x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -5, so the vertex is (2, -5).

Any non-zero number raised to the power of 0 equals 1.

The equation factors to (x + 2)² = 0. Therefore, x + 2 = 0, and x = -2.

The quotient rule states that when dividing powers with the same base, you subtract the exponents.

In the vertex form of a quadratic equation, y = a(x − h)² + k, the value 'k' represents the y-coordinate of the vertex of the parabola. The vertex is the point (h, k).

In vertex form, y = a(x - h)² + k, the vertex is (h, k). In this equation, h = 3 and k = 5, so the vertex is (3, 5).

A fractional exponent represents both a root and a power.

The power of a product rule: aᵐ × bᵐ = (ab)ᵐ.

This is the quotient rule: aᵐ/aⁿ = aᵐ⁻ⁿ.

The ± symbol in the quadratic formula indicates that there are two possible solutions (roots) for x. ¹ One solution is found by using the + sign, and the other is found by using the - sign

The power of a quotient rule states that both the numerator and denominator are raised to the power m.

The equation x² - 10x + 25 = 0 can be factored as (x - 5)² = 0. Taking the square root of both sides gives x - 5 = 0, therefore x = 5.

The power of a product rule states (ab)ᵐ = aᵐbᵐ.

To complete the square for an equation in the form x² + bx = c, you add (b/2)² to both sides. In this case, b = -6, so (b/2)² = (-6/2)² = (-3)² = 9. Therefore, you add 9 to both sides of the equation.

The discriminant D = b² - 4ac = 2² - 4 * 2 * 1 = 4 - 8 = -4. Since D < 0, the equation has two complex (non-real) roots.

The nth root of aᵐ is equivalent to a raised to the power of m/n.

In vertex form y = a(x - h)² + k, the vertex is (h, k). Here, the equation is y = -2(x - (-1))² + 3, so h = -1 and k = 3, giving the vertex (-1, 3).

x² + 6x + 9 is a perfect square trinomial and factors to (x + 3)(x + 3) = 0, which is equivalent to (x + 3)² = 0. Therefore, x = -3.

Any non-zero expression raised to the power of 0 equals 1.

Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a = (4 ± √((-4)² - 4 * 2 * 2)) / (2 * 2) = (4 ± √(16 - 16)) / 4 = (4 ± 0) / 4 = 1.

If D > 0, the quadratic equation has two distinct real roots.

The discriminant (D) of a quadratic equation in the form ax² + bx + c = 0 is calculated using the formula D = b² - 4ac.

In the given equation, 2x² + 3x + 1 = 0:

• a = 2
• b = 3
• c = 1

Therefore:

D = (3)² - 4 * 2 * 1 D = 9 - 8 D = 1

This is the power rule: (aᵐ)ⁿ = aᵐⁿ.

(√a)³ = (a¹/²)³ = a^(1/2 * 3) = a³/².

When dividing powers with the same base, subtract the exponents: aᵐ/aⁿ = aᵐ⁻ⁿ.