Linear / Quadratic Systems of Equations

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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.

Tags

CCSS.8.F.B.4

CCSS.HSF.LE.A.2

Front

What is a quadratic equation?

Back

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

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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.

Tags

CCSS.HSA.REI.C.7

Front

What does it mean if a linear-quadratic system has no solutions?

Back

It means that the graphs of the equations do not intersect at any point, indicating that there are no common solutions.

Tags

CCSS.HSA.REI.C.7

Front

What does it mean if a linear-quadratic system has one solution?

Back

It means that the graphs of the equations touch at exactly one point, indicating a single common solution.

Tags

CCSS.HSA.REI.C.7

Front

What does it mean if a linear-quadratic system has two solutions?

Back

It means that the graphs of the equations intersect at two distinct points, indicating two common solutions.

Tags

CCSS.HSA.REI.C.7

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.

Tags

CCSS.HSA-REI.B.4B

Front

How can you graph a quadratic function?

Back

To graph a quadratic function, find the vertex, axis of symmetry, and intercepts. Plot these points and draw a smooth curve through them.

Tags

CCSS.HSF-IF.C.7A

Front

What is the standard form of a linear equation?

Back

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