Linear algebra

A quadratic equation is a polynomial equation of the form:

ax2+bx+c=0

where a, b and care constants, and x represents the unknown variable. The key feature of a quadratic equation is that the highest power of x is 2.

Quadratic Formula

To find the solutions (or roots) of a quadratic equation, we can use the quadratic formula:

Factorization by Mid-term Splitting is a method used in algebra to factor quadratic expressions (or polynomials) of the form:

ax2+bx+c

where a, b, and c are constants. The goal is to express the quadratic expression as a product of two binomials. This technique is also referred to as factorization by splitting the middle term or splitting the linear term, because the middle term bx is split into two terms that facilitate factoring by grouping.

Completing the Square is a method used to solve quadratic equations, or to rewrite a quadratic expression in a more useful form. It involves manipulating the equation to form a perfect square trinomial, which can then be factored into the square of a binomial. This technique is often applied to solve quadratic equations or simplify expressions in various mathematical contexts.

Graphing a linear equation can be done by plotting points on a coordinate plane. One effective method to generate these points is by using a table of values. Here's the step-by-step process:

Choose several values for x (typically between -3 and 3), and substitute them into the equation to find corresponding values of y or vice -versa. This forms a table of (x, y)coordinate pairs.

Using the coordinates from the table, plot each point (x,y) on the coordinate plane.

Once the points are plotted, draw a straight line that passes through all of them.

1. Arrange the Equations: Write the system of equations in standard form Ax + By=C (if not already in that form), where A, B, and Care constants.

2. Align the Variables: Ensure that both equations are aligned, with corresponding variables and constants on opposite sides.

3. Multiply to Match Coefficients: If necessary, multiply one or both equations by a constant so that the coefficients of one of the variables (either x or y) are opposites. This makes it easier to eliminate one variable when the equations are added or subtracted.

4. Add or Subtract Equations: Add or subtract the equations to eliminate one of the variables, leaving a single variable in the resulting equation.

5. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.

6. Substitute Back: Once you have the value of one variable, substitute it into one of the original equations to solve for the other variable.

7. Check the Solution: Plug the values of both variables into the original equations to verify that they satisfy both equations.

1. Solve one equation for one variable:

   • Choose one of the equations and solve for either x or y (or whichever variables are used). This creates an expression for that variable in terms of the other variable.

2. Substitute the expression into the other equation:

   • Substitute the expression found in step 1 into the other equation. This will create a new equation with only one variable.

3. Solve for the remaining variable:

   • Solve this new equation for the remaining variable (usually straightforward algebra).

4. Substitute the value back:

   • Substitute the value of the solved variable back into the expression from step 1 to find the value of the other variable.

5. Check your solution (optional but recommended):

   • Plug the values of both variables back into the original equations to ensure that they satisfy both.

• Graph the Points: Plot the points on a Cartesian coordinate system for both the equations.

• Draw the Lines: Draw straight lines through the points for each equation, extending them across the graph.

1. Identifying the Point of Intersection:

   • The solution to the system of linear equations is represented by the point where the two lines intersect. This point corresponds to the values of x and y that satisfy both equations simultaneously.

2. Analyzing Possible Outcomes:

   • One Solution: The lines intersect at one point (as in the example).

   • No Solution: The lines are parallel and never intersect, indicating that the equations are inconsistent.

   • Infinite Solutions: The lines coincide (are the same line), meaning there are infinitely many solutions.

3. Verification:

   • It’s a good practice to substitute the intersection point back into the original equations to verify that it satisfies both equations.