How Many Times Can You Take the SAT? When Should 2025

The slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. Although y = b + mx is algebraically equivalent, y = mx + b is the conventional and widely accepted representation.

The coefficient of x is the number multiplied by x. In the equation 3x + 2y = 6, the number multiplied by x is 3.

Explanation: The slope-intercept form is y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis)

If the discriminant (D = b² - 4ac) is greater than zero, the quadratic equation has two different real solutions (roots).

The slope-intercept form of a linear equation is:

$$y = mx + b$$

Where:

• $m$ is the slope of the line.
• $b$ is the y-intercept.

In the equation $y = 2x - 5$, the coefficient of $x$ is $m = 2$, which represents the slope of the line.

Thus, the correct answer is A) 2.

In the factored form a(x − p)(x − q) = 0, p and q represent the roots (or solutions) of the quadratic equation. When x = p or x = q, the equation equals zero.

The slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept.

The expression (x + b/2a)² = (b² - 4ac) / 4a² is a key step in the process of deriving the quadratic formula by completing the square. It shows the result of rewriting the quadratic equation by creating a perfect square trinomial on one side. This is the core mechanism of the "completing the square" method. It's not the final quadratic formula itself, nor is it the vertex form or factored form, but rather a crucial intermediate step.

If D < 0, the roots are complex (no real roots). Complex roots come in conjugate pairs.

In the slope-intercept form (y = mx + b), 'm' represents the slope of the line

The coefficient of y is the number multiplied by y. In the equation 5x + 3y = 9, the number multiplied by y is 3.

A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

• a) x³ + 2x = 0 is a cubic equation (the highest power of x is 3).
• b) x² + 5x + 6 = 0 is a quadratic equation.
• c) 2x + 5 = 0 is a linear equation (the highest power of x is 1).
• d) x² - 5 = 0 is also a quadratic equation (in this case, b=0).

Explanation: The equation y − 4 = 2(x − 1) is in point-slope form. It represents a line with a slope of 2 that passes through the point (1, 4).

The discriminant, D = b² − 4ac, determines the nature of the roots (solutions) of a quadratic equation (ax² + bx + c = 0). Specifically:  

• If D > 0, the equation has two distinct real roots.  
• If D = 0, the equation has one real root (a repeated or double root).
• If D < 0, the equation has two complex roots (no real roots)

The standard form of a linear equation is Ax + By = C. To convert y = 4x + 5 to standard form, subtract 4x from both sides of the equation:

y - 4x = 4x + 5 - 4x -4x + y = 5

Multiplying both sides by -1 gives:

4x - y = -5

In a quadratic equation (ax² + bx + c = 0), the term ax² is called the quadratic term or the squared term. It's the term with the highest power of the variable x, which is x².

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

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 slope-intercept form is y = mx + b. Given m = -5 and b = 2, substituting these values into the equation gives y = (-5)x + 2, or y = -5x + 2.

If D = 0, the quadratic equation has exactly one real root (also called a repeated or double root).

In the slope-intercept form, bbb represents the y-intercept, where the line crosses the y-axis.

The general or standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Option b) represents a quadratic function.

In the slope-intercept form of a linear equation (y = mx + b), 'b' represents the y-intercept. The y-intercept is the point where the line intersects the y-axis.

The vertex form of a quadratic equation, y = a(x − h)² + k, directly reveals the vertex of the parabola, which is located at the point (h, k).

In the point-slope form of a linear equation, m represents the slope of the line.

Explanation: The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is usually non-negative.

A line with a slope of 5 and a y-intercept of 3 has the slope-intercept equation y = 5x + 3. To convert this to standard form, we subtract y from both sides and subtract 3 from both sides (or simply subtract y from both sides and then multiply both sides by -1):

y - y = 5x - y + 3 0 = 5x - y + 3 -3 = 5x - y 5x - y = -3

The standard form of a linear equation is Ax + By = C, where A, B, and C are constants (typically integers), and A is usually non-negative.

The expression b² - 4ac is called the discriminant. It determines the number and type of solutions (roots) a quadratic equation has:

• If b² - 4ac > 0, the equation has two distinct real roots.
• If b² - 4ac = 0, the equation has one real root (a repeated or double root).
• If b² - 4ac < 0, the equation has two complex roots (no real roots).

The standard form of a linear equation is:

$$Ax + By = C$$

Where:

• $A$, $B$, and $C$ are constants.
• The equation is in the form where both $x$ and $y$ terms are on the left side, and the constant is on the right side.

This form is often used because it is easy to apply when solving systems of linear equations.

Thus, the correct answer is C) $Ax + By = C$.

In the vertex form of a quadratic equation, y = a(x − h)² + k, the point (h, k) represents the vertex of the parabola.

Explanation: The equation x = 2 represents a vertical line where the x-coordinate is always 2, regardless of the y-value.

The point-slope form of a linear equation is:

$$y - y_1 = m(x - x_1)$$

Where:

• $m$ is the slope of the line.
• $(x_1, y_1)$ is a point on the line.

Given that the slope $m = 3$ and the point is $(2, 5)$, we substitute these values into the point-slope form:

$$y - 5 = 3(x - 2)$$

Thus, the correct equation is A) $y - 5 = 3(x - 2)$.

In the point-slope form of a linear equation, (x₁, y₁) represents a known point that the line passes through.

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. This form is used to find the solutions (roots or x-intercepts) of the quadratic. The form y = ax² + bx + c represents a quadratic function.

The slope-intercept form of a linear equation is:

$$y = mx + b$$

Where:

• $m$ is the slope of the line.
• $b$ is the y-intercept, the point where the line crosses the y-axis.

This is the standard form for expressing a linear equation in terms of its slope and y-intercept.

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).

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 point-slope form of a linear equation is:

$$y - y_1 = m(x - x_1)$$

Where:

• $m$ is the slope of the line.
• $(x_1, y_1)$ is a point on the line.

In the equation $y - 4 = -3(x - 2)$, the coefficient of $(x - 2)$ is $m = -3$, which represents the slope of the line.

Thus, the correct answer is B) -3.

When the discriminant (D) is greater than zero (D > 0), the quadratic equation has two different (distinct) real number solutions (roots). While option (b) says "two real roots" it is less precise than (c). If D=0 then there are two identical real roots. Thus, when D>0 we say there are two distinct real roots.

The standard form of a quadratic equation is ax² + bx + c = 0. The condition a ≠ 0 is crucial because if a were 0, the x² term would disappear, and the equation would become bx + c = 0, which is a linear equation, not a quadratic equation. Therefore, a ≠ 0 ensures that the equation is indeed quadratic because the x² term is present.

The process of rewriting the quadratic equation ax² + bx + c = 0 in the form a(x − p)(x − q) = 0 is called factoring. The values 'p' and 'q' represent the roots of the equation.

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. This formula provides the solutions (roots) for a quadratic equation in the standard form ax² + bx + c = 0.

The equation x² - 4 = 0 can be factored as a difference of squares: (x - 2)(x + 2) = 0. Setting each factor equal to zero gives the solutions:

• x - 2 = 0, which means x = 2
• x + 2 = 0, which means x = -2

Therefore, the roots are x = 2 and x = -2.

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.

To convert the equation from point-slope form to standard form (Ax + By = C), follow these steps:

1. Distribute: y - 3 = 2x - 2
2. Rearrange: Subtract 2x from both sides: -2x + y - 3 = -2
3. Add 3 to both sides: -2x + y = 1
4. Multiply by -1 (optional, but standard practice to have a positive A value): 2x - y = -1 * -1 which simplifies to 2x - y = 1

Therefore, the standard form is 2x - y = 1.

The standard form of a linear equation is:

$$Ax + By = C$$

Where:

• $A$, $B$, and $C$ are constants.
• $A$ is the coefficient of $x$.

In the equation $3x + 4y = 12$, the coefficient of $x$ is $A = 3$.

Thus, the correct answer is A) 3.