ACT Math Polynomials and Quadratic Equations Quiz

To solve the quadratic equation x² + 5x + 6 = 0, we can factor it into (x + 2)(x + 3) = 0. Setting each factor to zero gives the solutions x = -2 and x = -3. These values satisfy the original equation, confirming them as the correct answers.

To factor the polynomial 2x² + 7x + 3, we look for two numbers that multiply to 2 * 3 = 6 and add to 7. The numbers 6 and 1 fit this requirement. We can rewrite the middle term, allowing us to factor by grouping, resulting in (2x + 3)(x + 1).

To convert the quadratic equation y = x² - 4x + 5 into vertex form, complete the square. Rearranging gives y = (x² - 4x) + 5. Adding and subtracting 4 inside the parentheses yields y = (x - 2)² + 1, revealing the vertex at (2, 1) and confirming the correct vertex form.

To solve the quadratic equation x² - 6x + 8 = 0 using the quadratic formula, we identify coefficients a = 1, b = -6, and c = 8. Plugging these into the formula yields the roots x = 2 and x = 4, which are the points where the parabola intersects the x-axis.

Given that one root of the quadratic equation \(x^2 + bx + 12 = 0\) is 3, we can use Vieta's formulas. The sum of the roots is equal to \(-b\). If the other root is \(r\), then \(3 + r = -b\) and \(3r = 12\). Solving these gives \(r = 4\) and \(b = -7\).

To expand the expression (x + 4)(x - 2), apply the distributive property (FOIL method). Multiply the first terms (x*x), the outer terms (x*-2), the inner terms (4*x), and the last terms (4*-2). Combining these results gives x² - 2x + 4x - 8, which simplifies to x² + 2x - 8.

For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the sum of the roots is given by the formula \( -\frac{b}{a} \). Here, \( a = 1 \) and \( b = -8 \). Thus, the sum of the roots is \( -\frac{-8}{1} = 8 \).

To solve the equation 3x² - 12 = 0, first add 12 to both sides to get 3x² = 12. Then, divide by 3 to yield x² = 4. Taking the square root of both sides results in x = ±2, as both positive and negative roots are valid solutions for x².

To factor the expression x³ - 4x² + 4x, first, factor out the common term x, resulting in x(x² - 4x + 4). The quadratic can be further factored as (x - 2)². Thus, the complete factorization is x(x - 2)², which includes the linear factor and the perfect square.

To find the discriminant of the quadratic equation \(x² + 3x - 10 = 0\), use the formula \(D = b² - 4ac\). Here, \(a = 1\), \(b = 3\), and \(c = -10\). Calculating gives \(D = 3² - 4(1)(-10) = 9 + 40 = 49\). Thus, the discriminant is 49.

To find f(2), substitute x with 2 in the function f(x) = x² - 5x + 6. This gives f(2) = (2)² - 5(2) + 6 = 4 - 10 + 6 = 0. Thus, the value of f(2) is 0.

To complete the square for the equation x² + 8x = 0, we first add (8/2)² = 16 to both sides, transforming the left side into a perfect square trinomial. This results in (x + 4)² = 16, which represents the squared term equal to 16, making it easier to solve for x.