Exercises
Put your knowledge of quadratic equations to the test. This quiz covers solving equations for x, calculating discriminants, using the sum and product of roots, factoring quadratic expressions, and finding unknown coefficients from a given root. You will also practice quadratic inequalities, identify parabola properties such as the vertex and direction of opening, and rewrite equations by completing the square. Ideal for students building confidence with algebra, graphing, and essential quadratic formula concepts.
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To solve the quadratic equation x2 - 2x - 8 = 0, we can factor it to find the roots. The equation factors into (x - 4)(x + 2) = 0. Setting each factor equal to zero gives x - 4 = 0 or x + 2 = 0, which solve to x = 4 or x = -2. Therefore, the correct option is 2.
The discriminant of a quadratic equation of the form ax2 + bx + c = 0 is given by the formula b2 - 4ac. For the equation 3x2 - 6x + 2 = 0, a = 3, b = -6, and c = 2. Substituting these values, we get the discriminant as (-6)2 - 4(3)(2) = 12.
For a quadratic equation ax^2 + bx + c = 0, the sum of the roots can be found using the formula -b/a. In this equation, a = 1, b = -1, and c = -6. Therefore, the sum of the roots is -(-1)/1 = 1.
To factor x2 + x - 12, look for two numbers that multiply to -12 and add to 1. These numbers are 4 and -3. Therefore, the factorized form is (x + 4)(x - 3).
If 4 is a root of the quadratic equation x^2 + px + 16 = 0, then substituting x = 4 gives:
4^2 + 4p + 16 = 0
16 + 4p + 16 = 0
32 + 4p = 0
Solve for p:
4p = -32
p = -8
Therefore, the value of p is -8.
In a quadratic equation of the form ax^2 + bx + c = 0, the product of its roots is given by the formula c/a. For the equation x^2 - 5x + 6 = 0, a = 1 and c = 6. Therefore, the product of the roots is 6/1 = 6.
To solve the quadratic inequality x^2 - x - 6 > 0, we factor it: (x - 3)(x + 2) > 0. The critical points are x = 3 and x = -2. The inequality is true for x < -2 or x > 3. Thus, x = 0, which lies between -2 and 3, does not satisfy the inequality. Therefore, x = 0 is not a solution.
The vertex form of a parabolic equation is y = a(x - h)^2 + k, where (h, k) is the vertex. To find the vertex of y = 2x^2 - 4x + 1, use the formula for h: -b/(2a). Here, a = 2, b = -4. So, h = -(-4)/(2*2) = 1. Substitute x = 1 into the equation for y: 2(1)^2 - 4(1) + 1 = -1. Thus, the vertex is (1, -1).
A parabola represented by the equation y = ax^2 + bx + c opens downwards when the coefficient a is negative. In option 1, the equation is y = -x^2 + 4x - 5, where the coefficient a is -1, which is negative. Therefore, this equation represents a parabola that opens downwards.
To complete the square for the quadratic equation x^2 - 6x + 8 = 0, follow these steps:
1. Rewrite the equation as x^2 - 6x = -8.
2. Take half of the coefficient of x, which is -6, divide it by 2 to get -3, then square it to get 9.
3. Add and subtract 9 on the left side: (x^2 - 6x + 9) - 9 = -8.
4. This gives (x - 3)^2 = 1.
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