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Questions and Answers
Set them equal to each other
Set them equal to each other
x^2 - 5x + 7 = 2x + 1
Simplify into '= 0' format (like a standard Quadratic Equation)
Simplify into '= 0' format (like a standard Quadratic Equation)
x^2 - 7x + 6 = 0
Solve the Quadratic Equation
Solve the Quadratic Equation
x = 1 \text{ and } x = 6
The matching y values are (also see Graph)
The matching y values are (also see Graph)
What about a system of two equations where one equation is linear, and the other is quadratic
What about a system of two equations where one equation is linear, and the other is quadratic
Remember that the slope-intercept form of the equation for a line is y=mx+b , an______ the stan______ar______ form of the equation for a parabola with a vertical axis of symmetry is y=a x^2 +bx+c,  a≠0. To avoi______ confusion with the variables, let us write the linear equation as y=mx+______  where m is the slope an______ ______ is the y-intercept of the line.
Remember that the slope-intercept form of the equation for a line is y=mx+b , an______ the stan______ar______ form of the equation for a parabola with a vertical axis of symmetry is y=a x^2 +bx+c,  a≠0. To avoi______ confusion with the variables, let us write the linear equation as y=mx+______  where m is the slope an______ ______ is the y-intercept of the line.
We can use a version of the ______ method to solve systems of this type.
We can use a version of the ______ method to solve systems of this type.
To avoid confusion with the variables, let us write the linear equation as y=______ Â where m is the slope and d is the y-intercept of the line. Substitute the expression for y from the linear equation, in the quadratic equation. That is, substitute ______ Â for y in y=a x^2 +bx+c.
To avoid confusion with the variables, let us write the linear equation as y=______ Â where m is the slope and d is the y-intercept of the line. Substitute the expression for y from the linear equation, in the quadratic equation. That is, substitute ______ Â for y in y=a x^2 +bx+c.
Now, rewrite the new quadratic equation in standard form. Subtract mx+d  from both sides. ( mx+d )−( mx+d )=( a x^2 +bx+c )−( mx+d ) 0=______
Now, rewrite the new quadratic equation in standard form. Subtract mx+d  from both sides. ( mx+d )−( mx+d )=( a x^2 +bx+c )−( mx+d ) 0=______
Now we have a quadratic equation in one variable, the solution of which can be found using the quadratic formula. The solutions to the equation a x^2 +(b−m)x+(c−d)=0   will give the ______ of the points of intersection of the graphs of the line and the parabola. The corresponding y-coordinates can be found using the linear equation.
Now we have a quadratic equation in one variable, the solution of which can be found using the quadratic formula. The solutions to the equation a x^2 +(b−m)x+(c−d)=0   will give the ______ of the points of intersection of the graphs of the line and the parabola. The corresponding y-coordinates can be found using the linear equation.
Another way of solving the system is to ______ the two functions on the same coordinate plane and identify the points of intersection.
Another way of solving the system is to ______ the two functions on the same coordinate plane and identify the points of intersection.
A similar method can be used to find the intersection points of a line and a ______ or a line and an ellipse.
A similar method can be used to find the intersection points of a line and a ______ or a line and an ellipse.
Don't let the term 'imaginary' get in your way - there is nothing imaginary about them. They are just extensions of the ______ numbers, just like rational numbers (fractions) are an extension of the integers.
Don't let the term 'imaginary' get in your way - there is nothing imaginary about them. They are just extensions of the ______ numbers, just like rational numbers (fractions) are an extension of the integers.
The name 'imaginary number' was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term 'imaginary number' now means simply a ______ number with a real part equal to 0, that is, a number of the form bi.
The name 'imaginary number' was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term 'imaginary number' now means simply a ______ number with a real part equal to 0, that is, a number of the form bi.
The term 'imaginary number' now means simply a complex number with a real part equal to 0, that is, a number of the form ______.
The term 'imaginary number' now means simply a complex number with a real part equal to 0, that is, a number of the form ______.
The Real Numbers did not have a name before ______.
The Real Numbers did not have a name before ______.
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Study Notes
Solving Systems of Equations with Quadratic and Linear Equations
- To solve a system of two equations, one linear and one quadratic, we can use a version of the substitution method.
- The linear equation is in the slope-intercept form: y = mx + d, where m is the slope and d is the y-intercept of the line.
- The quadratic equation is in the standard form: y = ax^2 + bx + c, where a ≠0.
- Substitute the expression for y from the linear equation into the quadratic equation: ax^2 + bx + c = mx + d.
- Rewrite the new quadratic equation in standard form: 0 = ax^2 + (b - m)x + (c - d).
- Solve the quadratic equation using the quadratic formula to find the x-coordinates of the points of intersection.
- The corresponding y-coordinates can be found using the linear equation.
Imaginary Numbers
- Imaginary numbers are extensions of the real numbers, just like rational numbers (fractions) are an extension of the integers.
- The term 'imaginary number' was coined in the 17th century as a derogatory term, but now it means a complex number with a real part equal to 0.
- An imaginary number is in the form bi, where i is the imaginary unit and b is a real number.
- The Real Numbers did not have a name before the 17th century.
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