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.

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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.

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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.

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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.

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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.

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 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.

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.