Quadratic equations are polynomials of degree 2 and can be written in the form:

ax^{2} + bx + c = 0

Where a, b, and c are constants with 'a' not equal to 0.

The quadratic formula is used to solve quadratic equations. The formula is:

x = (-b ± √(b^{2} - 4ac)) / (2a)

- Identify the coefficients a, b, and c in the quadratic equation.
- Substitute the values of a, b, and c into the quadratic formula.
- Calculate the discriminant (the value inside the square root), which is b
^{2}- 4ac. - If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution. If it is negative, there are no real solutions (but there are complex solutions).
- Use the quadratic formula to calculate the values of x by plugging in the values of a, b, c, and the discriminant.

Let's solve the quadratic equation 2x^{2} + 5x - 3 = 0 using the quadratic formula.

Here, a = 2, b = 5, and c = -3.

Substitute these values into the quadratic formula:

x = (-5 ± √(5^{2} - 4*2*(-3))) / (2*2)

x = (-5 ± √(25 + 24)) / 4

x = (-5 ± √49) / 4

x = (-5 + 7) / 4 or x = (-5 - 7) / 4

So, the solutions are x = 1 or x = -3/2.

1. Solve the quadratic equation 3x^{2} - 4x + 1 = 0 using the quadratic formula.

2. Solve the quadratic equation 2x^{2} + 3x + 2 = 0 using the quadratic formula.

Remember to always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.