How to solve incomplete square equations?
Having learned to solve first-degree equations, you certainly want to work with others, in particular, with equations of the second degree, which are called quadratic in another way.
Square equations are equations of the type ax2 + bx + c = 0, where the variable is x, the numbers are - a, b, c, where a is not zero.
If in a quadratic equation one or another coefficient (c or b) is zero, then this equation will refer to an incomplete quadratic equation.
How to solve the incomplete quadratic equation, if the students still knew how to solve only the first-degree equations? Consider incomplete square equations of different types and simple ways to solve them.
a) If the coefficient c is 0, and the coefficient b is not zero, then ah2 + bx + 0 = 0 is reduced to an equation of the form ax2 + bx = 0.
To solve such an equation, we need to know the formulasolution of the incomplete quadratic equation, which consists in expanding the left side of the factor and then using the equality condition of the product to zero.
For example, 5x² - 20x = 0. We decompose the left side of the equation into multipliers, doing the usual mathematical operation: taking the common factor out of brackets
5x (x - 4) = 0
We use the condition that the products are zero.
5 x = 0 or x = 4 = 0
x = 0/5 x = 4
x = 0
The answer is: the first root is 0; the second root is 4.
b) If b = 0, and the free term is not zero, thenthe equation ax2 + 0x + c = 0 reduces to an equation of the form ax2 + c = 0. Solve the equations in two ways: a) expanding the polynomial of the equation in the left-hand side by multipliers; b) using the properties of an arithmetic square root. Such an equation is solved by one of the methods, for example:
4x² - 25 = 0
4x² = 25
x ² = 25/4
x = ± √ 25/4
x = ± 5/2. The answer is: the first root is 5/2; the second root is 5/2.
c) If b is 0 and c is 0, then ah ² + 0 + 0 = 0 is reduced to an equation of the form ax ² = 0. In such an equation, x will be 0.
As you can see, incomplete square equations can have no more than two roots.
Having learned to solve first-degree equations, you certainly want to work with others, in particular, with equations of the second degree, which are called quadratic in another way.
Square equations are equations of the type ax2 + bx + c = 0, where the variable is x, the numbers are - a, b, c, where a is not zero.
If in a quadratic equation one or another coefficient (c or b) is zero, then this equation will refer to an incomplete quadratic equation.
How to solve the incomplete quadratic equation, if the students still knew how to solve only the first-degree equations? Consider incomplete square equations of different types and simple ways to solve them.
a) If the coefficient c is 0, and the coefficient b is not zero, then ah2 + bx + 0 = 0 is reduced to an equation of the form ax2 + bx = 0.
To solve such an equation, we need to know the formulasolution of the incomplete quadratic equation, which consists in expanding the left side of the factor and then using the equality condition of the product to zero.
For example, 5x² - 20x = 0. We decompose the left side of the equation into multipliers, doing the usual mathematical operation: taking the common factor out of brackets
5x (x - 4) = 0
We use the condition that the products are zero.
5 x = 0 or x = 4 = 0
x = 0/5 x = 4
x = 0
The answer is: the first root is 0; the second root is 4.
b) If b = 0, and the free term is not zero, thenthe equation ax2 + 0x + c = 0 reduces to an equation of the form ax2 + c = 0. Solve the equations in two ways: a) expanding the polynomial of the equation in the left-hand side by multipliers; b) using the properties of an arithmetic square root. Such an equation is solved by one of the methods, for example:
4x² - 25 = 0
4x² = 25
x ² = 25/4
x = ± √ 25/4
x = ± 5/2. The answer is: the first root is 5/2; the second root is 5/2.
c) If b is 0 and c is 0, then ah ² + 0 + 0 = 0 is reduced to an equation of the form ax ² = 0. In such an equation, x will be 0.
As you can see, incomplete square equations can have no more than two roots.
