"Solving quadratic equations of one variable using formula method" PPT courseware

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"Solving quadratic equations of one variable using formula method" PPT courseware

This is how the formula method is produced

Can you solve the equation ax²+bx+c=0(a≠0) using the matching method?

1. Change to 1: Change the coefficient of the quadratic term to 1;

2. Move the term: Move the constant term to the right side of the equation;

3. Recipe: Add half the square of the absolute value of the coefficient of the linear term to both sides of the equation;

4. Deformation: factorize the left side of the equation and combine similar factors on the right side;

5. Square root: According to the meaning of square root, the square root of both sides of the equation;

6. Solve: Solve linear equations of one variable;

7. Definite solution: Write the solution of the original equation.

formula method

Generally, for the quadratic equation of one variable ax²+bx+c=0(a≠0)

The above formula is called the root finding formula of a quadratic equation of one variable.

The method of solving quadratic equations of one variable using the root formula is called the formula method

Teacher Tips:

The premise of using the formula method to solve quadratic equations of one variable is:

1. It must be a quadratic equation of general form: ax²+bx+c=0(a≠0).

2.b²-4ac≥0. It can be seen from the root finding formula that a quadratic equation of one variable has at most two real roots.

Example 1. Use the formula method to solve the equation 5x²-4x-12=0

1. Deformation: transform known equations into general forms;

2. Determine the coefficients: write the coefficients in terms of a, b, c;

3. Calculate: the value of b²-4ac;

4. Substitution: Substitute the relevant values ​​into the formula for calculation;

5. Definite roots: Write the roots of the original equation.

General steps for solving quadratic equations of one variable using the formula method:

1. Put the equation into general form and write the values ​​of a, b, and c.

2. Find the value of b²-4ac,

Special note: there is no solution when b²-4ac＜0

3. Substitute the root formula: ∴x=-b±√b²-4ac/2a

4. Write the solution to the equation: x1, x2

Practice in class

Solve the following equations using the formula method:

(1)2x2-9x+8=0;

(2)9x2+6x+1=0;

(3)16x2+8x=3.

There are three cases of roots of a quadratic equation (discriminant of roots)

1. When b²-4ac>0

The equation has two unequal real roots;

2. When b²-4ac=0

The equation has two equal real roots;

3. When b²-4ac＜0

The equation has no real roots;

Identify the roots of the following equations without solving the equations

1. x2-6x+1=0 has two unequal real roots

2. 2x2-x+2=0 has no real roots

3. 9x2+12x+4=0 has two equal real roots.

Thinking questions

1. When m takes any value, the equation x2+(2m+1)x+m2-4=0 has two equal real solutions.

2. The quadratic equation of x with respect to x is ax2+bx+c=0 (a≠0). When a, b, c satisfy what conditions, the two roots of the equation are opposite to each other?

Solve the following equations:

(1).x2-2x－8＝0;

(2).9x2+6x=8;

(3).(2x-1)(x-2) =-1;

Use your brain

The equation m2x2+(2m+1)x+1=0 about x has two unequal real roots, then m__________

Variation 1: The equation m2x2+(2m+1)x+1=0 about x has two equal real roots, then m_____________

Variation 2: The equation m2x2+(2m+1)x+1=0 about x has no real roots, then m____________

Variation 3: The equation m2x2+(2m+1)x+1=0 about x has two real roots, then m____________

Thought questions:

1. The quadratic equation of x with respect to x is ax2+bx+c=0 (a≠0). When a, b, c satisfy what conditions, the two roots of the equation are opposite to each other?

2. When m takes any value, the equation x2+(2m+1)x+m2-4=0 has two equal real number solutions.

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For more information about the PPT courseware "Using the Formula Method to Solve Quadratic Equations of One Dimension", please click the "Using the Formula Method to Solve Quadratic Equations of One Dimension" ppt tag.

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Update Time: 2024-11-19

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