Final answer:
The first step in solving Emily's equation uses the multiplication property of equality to eliminate the fraction. This technique and understanding the multiplication rules for signs are fundamental in equation solving, also applicable when using the quadratic formula or when completing the square.
Step-by-step explanation:
The original equation is given as: (1)/(4)(-3x²+2)+4=-x²+3. Emily's first step is to eliminate the fraction by multiplying both sides of the equation by 4, which results in: (-3x²+2)+16=-4x²+12. Here, multiplication property of equality is used because both sides of the equation are being multiplied by the same number, which does not change the equality. This property allows distribution of the multiplication across all terms within an equation.
When solving equations, it's essential to remember that multiplication or division by the same number on both sides does not affect the equation's equality. The multiplication rules for signs state when doing arithmetic operations like multiplication or division, the sign of the answer depends on the signs of the numbers involved.
Here's an example with multiplication involving different signs: 4 x (-4) = -16.
When dividing numbers, the same rules for signs apply, just as they do for multiplication. For division with opposite signs but equal magnitude, we might consider: x⁻¹ = ∕ implying that a negative exponent represents the reciprocal of the base raised to the positive of that exponent.
Lastly, solving quadratic equations such as x²+1.2 x 10⁻²x -6.0 × 10⁻³ = 0 would typically involve the quadratic formula or completing the square, allowing us to find the values of x.