How Do You Factor X^2 + 4X + 3?

     

Step by step solution :

Step 1 :

Step 2 :

Pulling out like terms :2.1 Pull out like factors:-x2 - 4x - 3=-1•(x2 + 4x + 3)

Trying lớn factor by splitting the middle term

2.2Factoring x2 + 4x + 3 The first term is, x2 its coefficient is 1.The middle term is, +4x its coefficient is 4.The last term, "the constant", is +3Step-1 : Multiply the coefficient of the first term by the constant 1•3=3Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is 4.

-3+-1=-4
-1+-3=-4
1+3=4That"s it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, 1 and 3x2 + 1x+3x + 3Step-4 : showroom up the first 2 terms, pulling out like factors:x•(x+1) add up the last 2 terms, pulling out common factors:3•(x+1) Step-5:Add up the four terms of step4:(x+3)•(x+1)Which is the desired factorization

Equation at the over of step 2 :

(-x - 3) • (x + 1) = 0

Step 3 :

Theory - Roots of a hàng hóa :3.1 A product of several terms equals zero.When a sản phẩm of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going khổng lồ solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.

Solving a Single Variable Equation:3.2Solve:-x-3 = 0Add 3 to lớn both sides of the equation:-x = 3 Multiply both sides of the equation by (-1) : x = -3

Solving a Single Variable Equation:3.3Solve:x+1 = 0Subtract 1 from both sides of the equation:x = -1

Supplement : Solving Quadratic Equation Directly

Solving x2+4x+3 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square & by using the Quadratic Formula

Parabola, Finding the Vertex:4.1Find the Vertex ofy = x2+4x+3Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up & accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,1, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can mã sản phẩm many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach.


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For this reason we want khổng lồ be able khổng lồ find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is -2.0000Plugging into the parabola formula -2.0000 for x we can calculate the y-coordinate:y = 1.0 * -2.00 * -2.00 + 4.0 * -2.00 + 3.0 or y = -1.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2+4x+3 Axis of Symmetry (dashed) x=-2.00 Vertex at x,y = -2.00,-1.00 x-Intercepts (Roots) : Root 1 at x,y = -3.00, 0.00 Root 2 at x,y = -1.00, 0.00

Solve Quadratic Equation by Completing The Square

4.2Solvingx2+4x+3 = 0 by Completing The Square.Subtract 3 from both side of the equation :x2+4x = -3Now the clever bit: Take the coefficient of x, which is 4, divide by two, giving 2, and finally square it giving 4Add 4 lớn both sides of the equation :On the right hand side we have:-3+4or, (-3/1)+(4/1)The common denominator of the two fractions is 1Adding (-3/1)+(4/1) gives 1/1So adding lớn both sides we finally get:x2+4x+4 = 1Adding 4 has completed the left hand side into a perfect square :x2+4x+4=(x+2)•(x+2)=(x+2)2 Things which are equal lớn the same thing are also equal to lớn one another. Sincex2+4x+4 = 1 andx2+4x+4 = (x+2)2 then, according lớn the law of transitivity,(x+2)2 = 1We"ll refer to lớn this Equation as Eq. #4.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+2)2 is(x+2)2/2=(x+2)1=x+2Now, applying the Square Root Principle khổng lồ Eq.#4.2.1 we get:x+2= √ 1 Subtract 2 from both sides khổng lồ obtain:x = -2 + √ 1 Since a square root has two values, one positive and the other negativex2 + 4x + 3 = 0has two solutions:x = -2 + √ 1 orx = -2 - √ 1

Solve Quadratic Equation using the Quadratic Formula

4.3Solvingx2+4x+3 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B và C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= 4C= 3 Accordingly,B2-4AC=16 - 12 =4Applying the quadratic formula : -4 ± √ 4 x=—————2Can √ 4 be simplified ?Yes!The prime factorization of 4is2•2 to lớn be able to remove something from under the radical, there have lớn be 2 instances of it (because we are taking a square i.e. Second root).√ 4 =√2•2 =±2 •√ 1 =±2 So now we are looking at:x=(-4±2)/2Two real solutions:x =(-4+√4)/2=-2+= -1.000 or:x =(-4-√4)/2=-2-= -3.000