Linear / Quadratic Systems of Equations

Front

What is a linear equation?

Back

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of the second degree, typically in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you find the intersection of a linear and a quadratic equation?

Back

To find the intersection, set the equations equal to each other and solve for x. Substitute the x-values back into either equation to find the corresponding y-values.

Front

What does it mean if a system of equations has no solution?

Back

A system has no solution if the lines represented by the equations are parallel and never intersect.

Front

What does it mean if a system of equations has one solution?

Back

A system has one solution if the lines represented by the equations intersect at exactly one point.

Front

What does it mean if a system of equations has two solutions?

Back

A system has two solutions if the quadratic equation intersects the linear equation at two distinct points.

Front

What is the vertex of a quadratic function?

Back

The vertex is the highest or lowest point of the parabola represented by the quadratic function, found at the point (h, k) in the vertex form y = a(x-h)² + k.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots: if positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.

Front

How can you solve a quadratic equation by factoring?

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To solve by factoring, express the quadratic in the form (x - p)(x - q) = 0, where p and q are the roots. Set each factor to zero and solve for x.

Front

What is substitution in solving systems of equations?

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Substitution involves solving one equation for one variable and then substituting that expression into the other equation.